On the Cartan decomposition for classical random matrix ensembles

Edelman, Alan; Jeong, Sungwoo

doi:10.1063/5.0087010

We complete Dyson’s dream by cementing the links between symmetric spaces and classical random matrix ensembles. Previous work has focused on a one-to-one correspondence between symmetric spaces and many but not all of the classical random matrix ensembles. This work shows that we can completely capture all of the classical random matrix ensembles from Cartan’s symmetric spaces through the use of alternative coordinate systems. In the end, we have to let go of the notion of a one-to-one correspondence. We emphasize that the KAK decomposition traditionally favored by mathematicians is merely one coordinate system on the symmetric space, albeit a beautiful one. However, other matrix factorizations, especially the generalized singular value decomposition from numerical linear algebra, reveal themselves to be perfectly valid coordinate systems that one symmetric space can lead to many classical random matrix theories. We establish the connection between this numerical linear algebra viewpoint and the theory of generalized Cartan decompositions. This, in turn, allows us to produce yet more random matrix theories from a single symmetric space. Yet, again, these random matrix theories arise from matrix factorizations, though ones that we are not aware have appeared in the literature.

I. INTRODUCTION

Random matrix theory (RMT) is a big subject touching so many fields of mathematics, science, and engineering. For such a subject, it is helpful to have a means of cataloging the objects to be studied and a theory that covers the objects in the catalog. In 1962, Dyson^1–4 was the first to propose a systematic approach to RMT. In the beginning of Ref. 4, he states his noble intent:

To bring together and unify three trends of thought which have grown up independently during the last thirty years.

which he enumerates as (i) group representations including time-inversion, (ii) Weyl’s theory of matrix algebras, and (iii) RMT.

Around a decade later, Dyson hit upon the idea that symmetric spaces should play a key role (Ref. 5, Sec. V). Dyson’s suggestion was taken up in famous papers by Zirnbauer et al.^6,7 and others.^8,9 These papers mainly focus on the noncompact cases. On the mathematical side, inspired by Katz and Sarnak,^10,11 Dueñez detailed connections to RMT for the compact symmetric spaces.^12,13

Nonetheless, we felt there was a gap. When one juxtaposes (i) the well-established theory of classical random matrix ensembles with (ii) the RMTs associated with symmetric spaces, ensembles are missing. In particular, only very special Jacobi ensembles (the left side of Fig. 2) seem to be making the symmetric space list. More precisely, if one starts with a symmetric space, one has to make what we call a coordinate system choice, what others might call a matrix factorization choice. This choice has been the map Φ : K × A → G/K; (k, a) ↦ kaK of Cartan, which we could call the KAK decomposition. (Although it is often called Cartan’s KAK decomposition, Cartan was not aware of G = KAK.) See Fig. 1.

FIG. 1.

View large Download slide

Families of matrix factorizations associated with a symmetric space, its tangent space, and its isometry group: Shown above are the skeleton of five factorizations associated with noncompact (left) and compact (right) symmetric spaces. Each serves as coordinate systems on the respective manifolds. Previous approaches (manifold, coordinate system, and measure) are shown in magenta. Examples of the linked factorizations/coordinate systems are shown.

We show that coordinate systems from the generalized Cartan (K₁AK₂) decomposition associate a single symmetric space to multiple RMTs. Letting go of the historical bias of the KAK decomposition, the full set of Jacobi ensembles (the right side of Fig. 2) emerges, thereby leading to the complete list of classical random matrix ensembles. Of course, there is much mathematical precedent in differential geometry to letting go of any one special coordinate system.

FIG. 2.

View large Download slide

The parameter space (α₁, α₂) ∈ (−1,∞)² of the β = 2 Jacobi ensemble obtained from Cartan’s coordinates (KAK) (left) and the generalized singular value decomposition coordinates (K₁AK₂) (right).

A. Classical random matrix ensembles

The objects that we are interested in are the classical random matrix ensembles. Well-established conventions in random matrix theory agree that the ensembles in this class consist of the Hermite, Laguerre, Jacobi, and circular ensembles built from matrices of integer sizes and involve entries that are real, complex, or quaternion. (Dyson denoted β = 1, 2, 4, and other authors in mathematics denote α = 2/β = 2, 1, 1/2.)

The term “classical random matrix ensembles” may be found in the following well-known references:

Chapter 1 of Forrester’s paper¹⁴ has the title “Classical Random Matrix Ensembles,” and the even sections (1.2, 1.4, 1.6, and 1.8) are explicitly Hermite, circular, Laguerre, and Jacobi in that order. (Odd sections have discussions related to these ensembles.) Forrester’s comprehensive book¹⁵ deals exclusively Hermite, Laguerre, Jacobi and circular ensembles in Chaps. 1–3 where the preface states: “eigenvalue p.d.f. of the various classical β-ensembles given in Chaps. 1–3.” Then, later in Chap. 5.4, he further justifies the terminology by pointing out the four weights from classical orthogonal polynomial theory.
In Ref. 16, Chap. 4.1 is entitled “Joint distribution of eigenvalues in the classical matrix ensembles” and specifically covers exactly the Hermite, Laguerre, Jacobi, and circular ensembles.
The first author’s 2005 Acta Numerica article (Ref. 17, Sec. 4).

If one starts with the list of ten infinite families of Cartan’s symmetric spaces (we will not discuss finite families of the exceptional types) and asks to characterize which classical random matrix ensembles are covered, answers could be found in Ref. 8 (Table 1), Ref. 9 (Table 1) (noncompact cases), and Ref. 13 (Table 1) (compact cases). However, turning the question around, if one starts with the classical random matrix ensembles and asks whether symmetric spaces are adequate to explain all of them, we find that the answer is a big “almost,” as the Jacobi ensembles are not adequately covered. To be precise, the Jacobi densities associated with compact symmetric spaces BDI, AIII, and CII from the previous attempts by the KAK decomposition are the following joint probability densities with β = 1, 2, 4 (up to constant) and integers p ≥ q,

KAK decomposition : \prod_{j < k} | x_{j} - x_{k} |^{β} \prod_{j = 1}^{q} x_{j}^{\frac{β}{2} - 1} {(1 - x_{j})}^{\frac{β (p - q + 1)}{2} - 1},

(1.1)

where we observe the powers of x_j’s restricted to $\frac{β}{2} - 1$ ⁠. The possible parameters of (1.1) are described in the left side of Fig. 2. Additional four compact symmetric spaces DIII, BD, C, and CI add four more Jacobi ensembles,¹³ but they are not sufficient to cover the two dimensional parameter set of the Jacobi ensembles.

B. Coordinate systems on the Grassmannian manifold

It is always interesting when a branch of applied mathematics reverses direction and provides guidance to pure mathematics. In this work, we focus on the role of the generalized singular value decomposition (GSVD) from numerical linear algebra.^18,19

From an applied viewpoint, the Jacobi ensembles are elegantly generated in software with commands such as svdvals(randn(p,s),randn(q,s)) in languages such as Julia, which is computed by taking the GSVD of two i.i.d. normal matrices with the same number of columns.^20,21 From a pure viewpoint, this is a pushforward of the uniform measure on the Grassmannian manifold onto a maximal Abelian subgroup A (with a fixed Weyl chamber) along the generalized Cartan (K₁AK₂) decomposition (Fig. 3).^22,23

FIG. 3.

View large Download slide

Cartan’s coordinate system (KAK) and GSVD coordinate systems (K₁AK₂) on the Grassmannian manifold O(n)/(O(n − s) × O(s)).

For example, take a Grassmannian point with any β = 1, 2, 4 from O(n)/(O(n − s) × O(s)) (respectively, with complex or quaternionic unitary groups) and represent it by the n × s orthogonal (respectively, complex or quaternionic unitary) matrix X. [More precisely, we treat the Grassmannian manifold as the quotient $V_{s} (R^{n}) / O (s)$ where $V_{s} (R^{n})$ is the Stiefel manifold. We are allowed to multiply any O ∈ O(s) on the right side of X.] For any p, q ≥ s satisfying p + q = n, we have the following coordinate system of X arising from the GSVD²⁴ of the first p rows and the last q rows of X (for an alternative viewpoint, see Ref. 25):

X = [\begin{matrix} U \\ V \end{matrix}] [\begin{matrix} C \\ S \end{matrix}] = [\begin{matrix} U C \\ V S \end{matrix}],

(1.2)

where U, V are p × s, q × s orthogonal (respectively, complex or quaternionic unitary) matrices and C, S are s × s diagonal matrices with cosine and sine values. Deduced joint probability densities²¹ (p, q ≥ s) are the following (up to constant):

K_{1} {AK}_{2} decomposition (GSVD) : \prod_{j < k} | x_{j} - x_{k} |^{β} \prod_{j = 1}^{s} x_{j}^{\frac{β (q - s + 1)}{2} - 1} {(1 - x_{j})}^{\frac{β (p - s + 1)}{2} - 1},

where the case q = s represents the usual KAK decomposition case (1.1).

As can be seen, the classical Jacobi parameters are quantized as they are integer multiples of β/2. Random matrix models that remove this quantization, thereby going beyond the classical, appear in Refs. 20, 26, and 27. In Sec. VII, we also illustrate that some Jacobi ensembles can arise from symmetric spaces that are outside the traditional quantization (Fig. 6).

C. Contributions of this paper

This work shows that a symmetric space can be associated with multiple random matrix theories (Fig. 4). Letting go of the arbitrariness of the choice of the KAK decomposition coordinate system allows us to choose other coordinate systems on symmetric spaces, thereby leading us to the complete list of classical random matrix ensembles (Secs. V, VI, VIII, and IX). Many of these coordinate systems are sometimes better known as matrix factorizations, used widely in matrix models of the classical ensembles.^{15,17,20,26,27} However, in Sec. VII, we compute new families of the Jacobi ensemble parameters from coordinate systems that have not been known before.

FIG. 4.

View large Download slide

Examples illustrating the lack of a one-to-one relationship between symmetric spaces and classical random matrix theories: A complex Grassmannian (top) obtains three Jacobi ensembles. A real Grassmannian (bottom) obtains four Jacobi ensembles. In particular, the β = 1 Jacobi ensemble $J_{0, 1}^{(1), 2}$ can be obtained from both symmetric spaces. Interestingly, a complex Grassmannian can lead to (top purple) a real RMT in the sense that β = 1. Similarly, a real Grassmannian obtains β = 2 RMT (bottom purple).

This work also endeavors to make the Lie theory more widely accessible by simplifying and modernizing key ideas and proofs in Ref. 28. Cartan’s theory^29–32 as developed by Helgason^28,33 is a crowning mathematical achievement, and it is our hope to open up this theory for the benefit of all. Indeed, in Ref. 34 (p. 428), Helgason writes about the difficulty of understanding Cartan’s writings:

[Cartan] was one of the great mathematicians of the period, but his papers were quite a challenge. Hermann Weyl, in reviewing a book by Cartan from 1937 writes: “Cartan is undoubtedly the greatest living master in differential geometry… I must admit that I found the book like most of Cartan’s papers, hard reading.”

In the same vein, while we are admirers of Helgason’s extensive work, we authors must admit that we, in turn, found Refs. 28 and 33 hard reading as well, and this paper attempts to introduce the theory by couching the ideas in terms of what we call ping pong operators.

Summarizing our work, we have the following:

We use the coordinate systems of the K₁AK₂ decomposition that connects a single symmetric space to multiple random matrices (Fig. 4), completing the list of associated classical random matrix ensembles.
We translate some of the key concepts in Cartan’s theory of symmetric spaces into easier to follow linear algebra (Sec. III).
We provide coordinate systems (matrix factorizations) of symmetric spaces that have not been discussed in random matrix context, obtaining new parameter families of the Jacobi ensemble (Sec. VII).

II. BACKGROUND

A. Joint densities of classical random matrix ensembles

Dyson introduced the β = 1, 2, 4 circular ensembles^1,4 in 1962. Earlier expositions on circular ensembles could be found on Hurwitz³⁵ and Weyl.³⁶ Hermite ensembles were introduced by Wigner.^37–39 Laguerre and Jacobi ensembles could be found as early as 1939 in the statistics literature by Fisher,⁴⁰ Roy,⁴¹ or Hsu.⁴² The physics literature first touches upon the idea of Laguerre and Jacobi with the 1963 thesis of Leff.⁴³ The following list is the joint probability densities (without normalization constants) of classical random matrix ensembles (β = 1, 2, 4):

Circular: $\prod_{j < k} | e^{i θ_{j}} - e^{i θ_{k}} |^{β}$ ⁠, $(θ_{1}, \dots, θ_{n}) \in [) 0, 2 π {()}^{n}$ ⁠;
Hermite: $\prod_{j < k} | λ_{j} - λ_{k} |^{β} e^{- \sum \frac{λ_{j}^{2}}{2}}$ $(λ_{1}, \dots, λ_{n}) \in R^{n}$ ⁠;
Laguerre: $\prod_{j < k} | λ_{j} - λ_{k} |^{β} \prod_{j = 1}^{m} λ_{j}^{α} e^{- \sum \frac{λ_{j}}{2}}$ $(λ_{1}, \dots, λ_{m}) \in [) 0, \infty {()}^{m}$ ⁠;
Jacobi: $\prod_{j < k} | x_{j} - x_{k} |^{β} \prod_{j = 1}^{m} x_{j}^{α_{1}} {(1 - x_{j})}^{α_{2}}$ (x₁, …, x_m) ∈ [0,1]^m.

In particular, the parameters α, α₁, α₂ > −1 are quantized as integer multiples of $\frac{β}{2}$ ⁠, i.e., $\frac{β}{2} (N + 1) - 1$ for some non-negative integer N.

B. Symmetric space and the generalized Cartan decomposition

In this section, we introduce the theory related to the generalized Cartan decomposition. For readers without preliminary knowledge in Lie theory, we recommend skipping to Sec. III, which follows a more modern linear algebra approach.

Let G/K_σ be a Riemannian symmetric space with a real reductive noncompact Lie group G and its maximal compact subgroup K_σ. Let σ be the Cartan involution on $g ≔ Lie (G)$ ⁠. Then, $g = k_{σ} + p_{σ}$ is the Cartan decomposition. Let τ be another involution on $g$ such that τσ = στ, and let $g = k_{τ} + p_{τ}$ be the ±1 eigenspace decomposition by τ. Denote by K_τ the analytic subgroup of G with tangent space $k_{τ}$ ⁠. Let $a$ be a maximal Abelian subalgebra of $p_{τ} \cap p_{σ}$ and define $A ≔ \exp (a)$ ⁠. We introduce the (noncompact) generalized Cartan decomposition (Ref. 22, Theorem 4.1).

Theorem 2.1

(generalized Cartan decomposition, K₁AK₂ decomposition). With the above setting, we have the following decomposition of G:

G = K_{τ} A K_{σ} .

(2.1)

That is, for any g ∈ G, we have k₁ ∈ K_τ, k₂ ∈ K_σ and a ∈ A such that g = k₁ak₂.

We often use the equivalent name “K₁AK₂ decomposition” for simplicity. Note that if τ = σ (i.e., K = K_σ = K_τ), we recover the usual KAK decomposition, G = KAK. The generalized Cartan decomposition in the work of Flensted-Jensen²² is originally intended for the case where G is noncompact. The compact analog is developed by Hoogenboom²³ [Theorem 3.6].

Theorem 2.2

(generalized Cartan decomposition; compact case). Let G/K_σ and G/K_τ be two compact Riemannian symmetric spaces. Let

g = k_{σ} + p_{σ}

and

g = k_{τ} + p_{τ}

be the corresponding eigenspace decompositions of

g = L i e (G)

. Then, for a maximal Abelian subalgebra

a

of

p_{σ} \cap p_{τ}

and

A = \exp (a)

⁠, we have

G = K_{τ} A K_{σ} .

From the space of linear functionals $a^{*}$ ⁠, we collect eigenvalues of an adjoint representation (the commutator) of $a$ on $g$ and call these eigenvalues the roots of the K₁AK₂ decomposition. By fixing the Weyl chamber, we obtain a set of positive roots Σ⁺. Details of the theory of the K₁AK₂ decomposition and its root system can be found in Flensted-Jensen,^22,44 Hoogenboom,²³ Matsuki,^45–47 and Kobayashi.⁴⁸ The K₁AK₂ decomposition is also studied in the context of spherical harmonics and intertwining functions.^49,50 Refine the root space $g_{α}$ of a root α by ±1 eigenspaces of στ. Let the two dimensions be $m_{α}^{\pm}$ ⁠.

Let dk_σ, dk_τ be the Haar measures of K_σ, K_τ, respectively. Let dH be the Euclidean measure on $a$ ⁠. The Jacobian of the K₁AK₂ decomposition is the following:

Theorem 2.3

(Jacobian of the K₁AK₂ decomposition^23,44). Let dg be the Haar measure on G, and let

H \in a

. We have the Jacobian and the integral formula corresponding to the change of variables associated with the K₁AK₂ decomposition,

\int_{G} f (g) d g = \int_{K_{τ}} \int_{K_{σ}} \int_{a^{+}} f (k_{σ} \exp (H) k_{τ}) d μ (H) d k_{σ} d k_{τ},

where for noncompact G,

d μ (H) \propto \prod_{α \in Σ^{+}} {(\sinh α (H))}^{m_{α}^{+}} {(\cosh α (H))}^{m_{α}^{-}} d H,

(2.2)

and for compact G,

d μ (H) \propto \prod_{α \in Σ^{+}} {(\sin α (H))}^{m_{α}^{+}} {(\cos α (H))}^{m_{α}^{-}} d H .

(2.3)

Similar results on the KAK decomposition and the restricted roots of symmetric spaces can be found in standard Lie group textbooks.^{28,33,51–53} In the KAK case, the Jacobian (2.2) reduces down to $\prod {(\sinh α (H))}^{m_{α}}$ as we do not have −1 eigenspace of στ so that $m_{α} = m_{α}^{+}$ ⁠.^33,54,55

Theorems 2.1–2.3 are decompositions of the group G. These decompositions can also be applied to the symmetric space G/K_σ. The following map Φ is the K₁AK₂ decomposition of the Riemannian symmetric space G/K_σ. The map Φ is also called the Hermann action,^56,57 nonstandard polar coordinates,⁵⁸ and non-Cartan parameterization.⁵⁹ In the KAK case (K = K_σ = K_τ), Helgason called this the polar coordinate decomposition³³ and credits Cartan³⁰ for this map. Since the G-invariant measure of G/K inherits the Haar measure of G, the identical Jacobian is obtained for the decomposition of a symmetric space.⁶⁰

Theorem 2.4

(K₁AK₂ decomposition of G/K_σ). Given a K₁AK₂ decomposition G = K_σAK_τ with the Riemannian symmetric space G/K_σ, we have the map Φ,

Φ : K_{τ} \times A \to G / K; (k_{τ}, a) \mapsto k_{τ} a K .

(2.4)

Suppose

H \in a

, a = exp(H). For the G-invariant measure dx of G/K_σ, dk_τ = Haar(K_τ), and the Euclidean measure dH on

a

, dx = dk_τdμ(H) holds where the Jacobian dμ(H) is given in (2.2) if G is noncompact and (2.3) if G is compact.

Remark 2.5

[representing G/K ≅ P: gK (coset) or p ∈ P?]. In the standard KAK decomposition, the Jacobian () [respectively, ()] only has sinh (respectively, sin) terms as we discussed above. This result could be found in many literature, where some authors^28,44,55,61 use ∏sinh α(H) as the Jacobian, whereas other authors^13,54,62 use ∏ sinh(α(H)/2). This gap is due to the difference in the realization of a symmetric space G/K as a subset P ⊂ G. The former uses the right coset representative, i.e., G/K → P as gK ↦ p, where g = pk is its group level Cartan decomposition. Then, the action of G on G/K is given as (g₁, g₂K) ↦ g₁g₂K. The latter authors use the map G/K → P such that gK ↦ g(σg)⁻¹, where σ is the group level involution. The G-action is

(g_{1}, g_{2}) \mapsto g_{1} g_{2} {(σ g_{1})}^{- 1}, g_{1} \in G, g_{2} \in P

⁠. In terms of Theorem 2.4, the latter gives the map Φ such that (k, a) ↦ ka²k⁻¹ since

g {(σ g)}^{- 1} = p k σ {(p k)}^{- 1} = p k {(p^{- 1} k)}^{- 1} = p k k^{- 1} p = p^{2} = k a k^{- 1} k a k^{- 1} = k a^{2} k^{- 1},

which explains the extra factor

\frac{1}{2}

applied to H where a = exp(H). Moreover, these two identifications define the map Φ : K × A → P with the same k, a as

Φ : (k, a) \mapsto k a K or Φ : (k, a) \mapsto k a^{2} k^{- 1},

(2.5)

depending on the author’s notational choice explained above. This coordinate system Φ is sometimes called the polar coordinate decomposition, e.g., see Ref. 33 (p. 402).

Example 2.6

(G/K vs P: a symmetric positive definite matrix). Let us take a look at the two realizations in Remark 2.5 for $G / K = GL (n, R) / O (n)$ ⁠, where P is the set of all symmetric positive definite matrices. Let S be a fixed positive definite symmetric matrix, with its eigendecomposition S = QΛQ^T, with Q ∈ O(n). The coset representation of S is QΛ · O(n) ∈ G/K as QΛ = (QΛQ^T)Q is the polar decomposition. With the realization of P ≅ G/K, the point in G/K is represented by the matrix S = QΛQ^T.

Finally, we have the Lie algebra counterpart of Theorem 2.4 when K = K_σ = K_τ.

Theorem 2.7.

For a noncompact Riemannian symmetric space G/K with the Cartan decomposition

g = k + p

⁠, let

a

be a maximal Abelian subalgebra of

p

. We have

Ψ : K \times a \to p; (k, H) \mapsto k H k^{- 1},

(2.6)

equivalently the decomposition

p = \cup_{k \in K} k a k^{- 1}

with the Jacobian dμ given as

d μ (H) \propto \prod_{α \in Σ^{+}} | α (H) |^{m_{α}},

(2.7)

where

H \in a

and Σ is the restricted root system with dimensions m_α. The measure on

p

is the Euclidean measure.

C. A symmetric space: one RMT or many RMTs?

The answer to the title question of this section is that both one and many can be construed as correct. To explain how this is possible requires teasing apart the assumptions behind the words “associated with.” Certainly,^6,8,9,13 associate one random matrix with one symmetric space. However, the example of the GSVD coordinate systems discussed in Sec. I B associates multiple Jacobi densities with one symmetric space, the Grassmannian manifold. In Ref. 59, another example is illustrated as the “non-Cartan parameterization” for the special case of (G, K_σ, K_τ) = (U(n), O(n), U(p) × U(q)). (A similar approach may be found in Ref. 63.) This is discussed in Sec. VII B.

The reconciliation is that indeed it is true that the required maps (2.4) when K = K_σ = K_τ, i.e., Φ(k, a) = kaK = kak⁻¹ (compact) or the map (2.6) ψ(k, H) = kHk⁻¹ (noncompact) lead to a unique random matrix theory associated with a given symmetric space G/K. This is unique in a sense that any geodesic on the symmetric space G/K could be transformed to the geodesic on A with the above maps.

However, if we relax the condition so that we are allowed to choose K_τ under the generalized Cartan decomposition framework, we can associate multiple random matrix theories to one symmetric space. The GSVD coordinate systems in Sec. I B illustrate this viewpoint. The real Grassmannian manifold G/K = O(n)/(O(n − s) × O(s)) has the map Φ: (k, a) ↦ kaK for K = K_σ = K_τ explicitly written as $X = [\begin{matrix} U \\ V \end{matrix}] [\begin{matrix} C \\ S \end{matrix}] \cdot O (s)$ ⁠, where U, V are (r − s) × s, s × s orthogonal matrices. On the other hand, if we let K_τ = O(p) × O(q), we have multiple maps Φ: (k_τ, a) ↦ k_τaK written as $X = [\begin{matrix} U \\ V \end{matrix}] [\begin{matrix} C \\ S \end{matrix}] \cdot O (s)$ ⁠, where U, V are p × s, q × s orthogonal matrices.

Starting from Sec. V, we discuss (i) random matrices arising from the K₁AK₂ decompositions of compact symmetric spaces (Theorem 2.4 or 2.2) and (ii) random matrices arising from the Lie algebra decomposition of noncompact symmetric spaces (Theorem 2.7). The associated decompositions are well explained by matrix factorizations in numerical linear algebra. As we pointed out, the resulting Jacobi ensembles cover the full parameter set of the classical Jacobi densities, thereby completing the classification from the classical RMT point of view.

III. CARTAN’S IDEA: A MODERNIZED APPROACH

The Jacobian of the KAK (K₁AK₂) decomposition, equivalently the determinant of the differential of the map Φ : K × A → P (in Theorem 2.4 and Remark 2.5), is computed in several references.^28,54,55 The proof of (2.2) is can also be found in Refs. 23 and 44. However, the proof can be inaccessible to some audiences. Meanwhile, individual cases of the KAK decomposition, recognized as matrix factorizations, show up in many areas of mathematics, and some were discovered in various formats by specialists in numerical linear algebra. Motivated by random matrix theory (and sometimes perturbation theory in numerical analysis), Jacobians of these factorizations were often computed case-by-case using the matrix differentials and wedging of independent elements.^{15,21,26,64,73}

In this section, we provide a generalization of such individual Jacobian computations and compare it to the general technique Helgason proposed. With appropriate translation of terminologies and maps in Lie theory into linear algebra, we observe the both methods are indeed the same process but have been illustrated in different languages for a long time. We start out by introducing some important concepts in Lie theory accessible to an audience with a good background in linear algebra and perhaps some basic geometry. Then, in Table II, we present a line-by-line correspondence between Helgason’s derivation and the proof by matrix differentials.

A. The ping pong operator, ping pong vectors, and ping pong subspaces

We will start with a concrete 2 × 2 linear operator so as to establish the notions of the ping pong operator, ping pong vectors, ping pong subspaces, and the relationship to eigenvectors. Then, we will define a “bigger” linear operator ad_H that acts on 2 × 2 spaces exactly in the manner we are about to describe.

We introduce the 2 × 2 matrix

M ≔ [\begin{matrix} 0 & α \\ α & 0 \end{matrix}] = α [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}],

which we will call a 2 × 2 ping pong operator, and we will call $[\begin{matrix} 1 \\ 0 \end{matrix}]$ and $[\begin{matrix} 0 \\ 1 \end{matrix}]$ the ping pong vectors of M, in that M bounces these two vectors into α times the other,

M [\begin{matrix} 1 \\ 0 \end{matrix}] = α [\begin{matrix} 0 \\ 1 \end{matrix}], M [\begin{matrix} 0 \\ 1 \end{matrix}] = α [\begin{matrix} 1 \\ 0 \end{matrix}] .

Furthermore, M has eigenvectors $[\begin{matrix} 1 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ - 1 \end{matrix}]$ ⁠, with eigenvalues α, −α. We will call the eigenvalue a root of M.

Also worth pointing out are the matrix exponential and matrix sinh of M,

e^{M} = [\begin{matrix} \cosh α & \sinh α \\ \sinh α & \cosh α \end{matrix}] and \sinh M = \frac{1}{2} (e^{M} + e^{- M}) = \sinh α \cdot [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}],

and thus, we see that sinh M is another ping pong operator with scaling sinh α. Figure 5 plots the action of a ping pong matrix and its exponential, with notations that we will use in Secs. III D and III E, i.e., the ping pong operator is denoted ad_H, p_j and k_j are the ping pong vectors, and x_j and θx_j are the eigenvectors. The right side of Fig. 5 shows the action of e^M and portrays sinh(M) as a projection of e^M on the p_j direction.

FIG. 5.

View large Download slide

The eigenmatrices x_j, θx_j and ping pong matrices k_j, p_j (3.4) in the tangent space $g$ ⁠. The operators are illustrated in blue lines. The operator ad_H and ping pong relationship (left) and the operator $e^{{a d}_{H}}$ on k_j to p_j (right). The left map shows the factor of α_j, which is a building block of the Jacobian ∏_j |α_j(H)| (2.7). The factor of sinh α_j in the right map builds the Jacobian ∏_j | sinh α_j(H)| (2.2).

We now go beyond 2 × 2 matrices and suggest the more general 2N × 2N ping pong matrix M_N, with N roots, α₁, …, α_N, N pairs of ping pong vectors (k₁, p₁), …, (k_N, p_N) along with eigenvectors (x₁, y₁), …, (x_N, y_N),

M_{N} = [\begin{matrix} 0 & α_{1} \\ α_{1} & 0 \\ ⋱ \\ 0 & α_{N} \\ α_{N} & 0 \end{matrix}],

k_{j}, p_{j}, x_{j}, y_{j} = [\begin{matrix} ⋮ \\ 1 \\ 0 \\ ⋮ \end{matrix}], [\begin{matrix} ⋮ \\ 0 \\ 1 \\ ⋮ \end{matrix}], [\begin{matrix} ⋮ \\ 1 \\ 1 \\ ⋮ \end{matrix}], [\begin{matrix} ⋮ \\ 1 \\ - 1 \\ ⋮ \end{matrix}], j = 1, 2, \dots, N,

where the 2j − 1 and 2j positions are 0 or ±1 and all other entries of these vectors are 0. The matrices exp(M_N) and sinh M_N are block versions of the 2 × 2 case.

We may define the subspaces $k$ and $p$ (using the “mathfrak” Fraktur letters “k” and “p”) to be the span of the k_j and p_j, respectively. Note that $k$ and $p$ are orthogonal complements as subspaces. A key “ping pong” relationship between these subspaces is that

\begin{matrix} M_{N} k \in p if k \in k, \\ M_{N} p \in k if p \in p . \end{matrix}

Thus, if we consider $M_{N} |_{k}$ ⁠, the restriction of M_N to $k$ ⁠, we have an operator from $k$ to $p$ ⁠. Evidently, $M_{N} |_{k}$ as a matrix may be obtained by taking the even rows and odd columns of M_N. The result is a diagonal matrix with the α_j on the diagonal. Similarly, $\sinh (M_{N}) |_{k}$ is a diagonal matrix with sinh(α_j) on the diagonal. We then get the important result that

\det (\sinh (M_{N}) |_{k}) = \prod_{j = 1}^{N} \sinh α_{j},

the product of the hyperbolic sines of the roots.

Given a linear operator $L$ on a vector space with nonzero eigenvalues ±λ, the following lemma constructs a pair of ping pong vectors from $L$ ⁠:

Lemma 3.1.

For a linear operator

L

defined on any vector space, assume that ±λ are both nonzero eigenvalues of

L

. Let x and y be the corresponding eigenvectors, i.e.,

L x = λ x

and

L y = - λ y

. Define two vectors k ≔ x + y, p ≔ x − y. Then, k, p are ping pong vectors. Furthermore, we have for the operator

\exp (L)

,

e^{L} k = \cosh λ k + \sinh λ p, e^{L} p = \sinh λ k + \cosh λ p .

The proof is a straightforward extension of the discussion in previous paragraphs.

Remark 3.2.

For the reader who wants to know the upcoming significance of this fact for Jacobians of matrix factorizations, it turns out (or maybe as the reader already observed in Sec. II) that the Jacobian will be the product of sinh α’s. Just as the matrix $\sinh ([\begin{matrix} 0 & α \\ α & 0 \end{matrix}])$ takes one of the ping pong vectors to sinh α times the other, the key piece of the differential map will consist of multiple ping pong relationships, each one sending one ping pong vector to another.

B. The Kronecker product, linear operator ad_X, and its exponential

Lie theory picks out operators $L$ that exactly have the properties in Sec. III A. Our vector spaces are now matrix spaces, and our operators are linear operators on a matrix space. We introduce the Lie bracket, denoted by [X, Y], defined as [X, Y] = XY − YX (the commutator). The Kronecker product notation is very helpful in this context. We define the Kronecker product notation as a linear operator on a matrix space. [Many authors would write vec(BXA^T) = (A ⊗ B)vec(X), but we omit the “vec” as we believe it is always clear from context. In a computer language such as Julia, one would write kron(A,B) * vec(X) = vec(B*X*A′)],

(A \otimes B) X = B X A^{T} .

(3.1)

With this, we can express the Lie bracket with Kronecker products,

(I \otimes X - X^{T} \otimes I) Y = X Y - Y X .

Consider the Lie bracket as a linear operator (determined by X) applied to Y, and call this operator ad_X (abbreviation for “adjoint”),

\begin{matrix} {a d}_{X} = I \otimes X - X^{T} \otimes I, \\ {a d}_{X} (Y) = [X, Y] . \end{matrix}

This will be the important ping pong operator $L$ ⁠. The operator exponential of ad_X (equivalently, the matrix exponential of I ⊗ X − X^T ⊗ I) is given in the following lemma:

Lemma 3.3.

For the linear operator ad_X, the following holds for

e^{{a d}_{X}} ≔ \sum_{j = 0}^{\infty} \frac{{({a d}_{X})}^{n}}{n!}

and

\sinh {a d}_{x} = (e^{{a d}_{X}} + e^{- {a d}_{X}}) / 2

:

e^{{a d}_{X}} = \exp (I \otimes X - X^{T} \otimes I) = {(e^{- X})}^{T} \otimes e^{X},

(3.2)

e^{{a d}_{X}} Y = e^{X} Y e^{- X} and (\sinh {a d}_{x}) Y = (e^{X} Y e^{- X} - e^{- X} Y e^{X}) / 2 .

(3.3)

Proof.

The proof is straightforward by identity ().

e^{X} Y e^{- X} = ({(e^{- X})}^{T} \otimes e^{X}) Y

and

e^{{a d}_{X}} Y = \exp (I \otimes X - X^{T} \otimes I) Y

⁠. It is left to prove

{(e^{- X})}^{T} \otimes e^{X} = \exp (I \otimes X - X^{T} \otimes I)

⁠. Since I ⊗ X commutes with X^T ⊗ I, we have

\exp (I \otimes X - X^{T} \otimes I) = e^{I \otimes X} e^{- X^{T} \otimes I} = (I \otimes e^{X}) ({(e^{- X})}^{T} \otimes I) = {(e^{- X})}^{T} \otimes e^{X},

proving the result. The sinh result follows trivially.□

C. Antisymmetric and symmetric matrices: An important first example of symmetric space as ping pong spaces

In our first example, our vector space is n × n real matrices. Consider

\begin{array}{l} k & = {Antisymmetric matrices}, \\ p = {Symmetric matrices} . \end{array}

The ping pong operator that will bounce $k$ and $p$ around will be ad_H = I ⊗ H − H^T ⊗ I, where H is the diagonal matrix

H = [\begin{matrix} h_{1} \\ ⋱ \\ h_{n} \end{matrix}] .

Note that the operator ad_H sends an antisymmetric matrix to a symmetric matrix and a symmetric matrix to an antisymmetric matrix.

What does this have to do with Jacobians of matrix factorizations, such as the symmetric positive definite eigenvalue factorization? Consider a perturbation of Q when forming S = QΛQ^T. An infinitesimal antisymmetric perturbation Q^TdQ is mapped into a dS, an infinitesimal symmetric perturbation. This is the very linear map from the tangent space of Q to that of S that we wish to understand, so perhaps it is not surprising we would want to restrict our ping pong operator from $k$ to $p$ ⁠. We invite the reader to check that the corresponding eigenmatrices and ping pong matrices of ad_H may be found in the first column of Table I.

TABLE I.

Examples of eigenmatrices x_l, θx_l and ping pong matrices k_l, p_l. k_l = x_l + θx_l and p_l = x_l − θx_l as defined in (3.4). k_l, p_l are normalized to have ±1 entries. A block structure on row/columns j, k and j′ ≔ p + j and k′ ≔ p + k are filled up with 0 and ±1.

$\frac{G}{K}$	$\frac{U (n)}{O (n)}$	$\frac{O (n)}{O (p) \times O (q)}$
x_l	⋯	⋯
θx_l	⋯	⋯
k_l
p_l

D. General $k$ and $p$ arise from an involution $θ$

We proceed to construct more important general operators $L$ that have the property in the assumption of Lemma 3.1. This is where the theory of Lie groups and symmetric spaces need to be brought in. Upon doing so, we will obtain two linear spaces of matrices $k$ ⁠, $p$ ⁠, and also a space $a$ ⁠.

For the reader not familiar with Lie groups, one need only imagine a continuous set of matrices that are a subgroup of real, complex, or quaternion matrices. The tangent space $g$ is just a vector space of matrix differentials at the identity. One key example is the compact Lie group O(n) (the group of square orthogonal matrices) and its tangent space at the identity $g_{O (n)}$ ⁠: the set of antisymmetric matrices. Another key example is all n × n invertible matrices $GL (n, R)$ (a noncompact Lie group) and its tangent space $g_{GL (n, R)}$ ⁠, consisting of all n × n matrices.

Cartan noticed that important matrix factorizations start with two ingredients: the tangent space $g$ (at the identity) of a Lie group G and an involution θ on $g$ (i.e., θ² = Id and θ[X, Y] = [θX, θY]) An example of θ is θ(X) = −X^T on $g$ for $G = GL (n, R)$ ⁠. Among matrices in $g$ ⁠, we select two kinds of matrices. The ones fixed by the involution θ, and the ones negated by θ. Denote each set by $k$ and $p$ ⁠,

k ≔ {g \in g : θ (g) = g}, p ≔ {g \in g : θ (g) = - g} .

[For $GL (n, R)$ ⁠, these are the antisymmetric and symmetric matrices respectively.]

The next important player is $a \subset p$ ⁠. Readers familiar with the singular value decomposition know the special role of diagonal matrices in the SVD as they list the very important “singular values.” Diagonal matrices have the nice property that linear combinations are still diagonal, they commute (the Lie bracket of any two are zero), and they are symmetric (the $p$ of our first example). The generalization of this is to take a $p$ and find a maximal subalgebra where every matrix commutes. This is the maximal subspace $a \subset p$ such that for all $a_{1}, a_{2} \in a$ ⁠, [a₁, a₂] = 0.

If $H \in a$ ⁠, then S = QΛQ^T is a symmetric positive definite eigendecomposition, with Λ = e^H. In the rest of the section, we will be focusing on factorizations of the form QΛQ⁻¹, where Λ is a matrix exponential of $H \in a$ ⁠. (These will be more general than eigendecompositions, as Q may not be orthogonal, and Λ may not be diagonal.) In particular, we will compute the Jacobian of perturbations with respect to Q, holding H constant, and thus, necessarily the Jacobian will be defined in terms of H.

From here, we assume that the Lie group G is noncompact. The compact case will be discussed after completing the noncompact case. Pick $H \in a$ ⁠, and recall that ad_H is a linear operator on $g$ ⁠. The operator ad_H will play the role of $L$ ⁠, the ping pong operator. We decompose $g$ into the eigenspaces of ad_H. For any eigenpair (α_j, x_j) of ad_H, i.e., ad_H(x_j) = [H, x_j] = α_jx_j, we observe (for α_j ≠ 0)

{a d}_{H} (θ x_{j}) = [H, θ x_{j}] = - [- H, θ x_{j}] = - [θ H, θ x_{j}] = - θ ([H, x_{j}]) = - α_{j} θ x_{j},

which implies that the eigenvalues ±α_j always exist in pairs, with the corresponding eigenmatrices x_j and θx_j. This satisfies the assumption of Lemma 3.1, from which we can now construct our ping pong matrices,

k_{j} ≔ x_{j} + θ x_{j}, p_{j} ≔ x_{j} - θ x_{j},

(3.4)

with the ping pong relationship by the operator ad_H,

{a d}_{H} k_{j} = α_{j} p_{j}, {a d}_{H} p_{j} = α_{j} k_{j} .

(3.5)

In addition, the relationship by the operator $e^{{a d}_{H}}$ follows:

e^{{a d}_{H}} k_{j} = \cosh α_{j} k_{j} + \sinh α_{j} p_{j},

(3.6)

e^{{a d}_{H}} p_{j} = \sinh α_{j} k_{j} + \cosh α_{j} p_{j} .

(3.7)

The ping pong matrices k_j, p_j, eigenmatrices x_j, θx_j and the relationships (3.5), (3.6) are illustrated in Fig. 5.

As we mentioned in Remark 3.2 and Sec. III C, the role of ping pong matrices k_j, p_j is crucial. The map $e^{{a d}_{H}}$ (particularly, sinh ad_H) is the main ingredient constructing the differential map dΦ of the factorization Φ: (Q, Λ) ↦ QΛQ⁻¹. The operator $e^{{a d}_{H}}$ is applied to k_j and then projected to the span of p_j as in Fig. 5 (right), leaving only the sinh α_j factor.

We now compute the full basis of $k$ and $p$ ⁠. The collection ∪_j{x_j, θx_j} is a full basis for the union of eigenspaces with nonzero eigenvalues. Since span({x_j, θx_j}) = span({k_j, p_j}) for any j, ∪_j{k_j, p_j} is another full basis for the eigenspaces with nonzero eigenvalues. Interestingly, we observe θk_j = k_j and θp_j = −p_j, which identifies ∪_j{k_j} and ∪_j{p_j} as subsets of the basis of $k$ and $p$ ⁠, respectively. The remaining case is the zero eigenspace. When α_j = 0, there are two possibilities. First, if x_j and θx_j are independent of each other, we can still obtain k_j and p_j as before and add them to ∪_j{k_j} and ∪_j{p_j}. Second, if x_j and θx_j are colinear, θx_j is either x_j or −x_j. If θx_j = x_j, we collect such x_j and name the set K_z. Similarly, if θx_j = −x_j, then we put them in P_z. Since we analyzed both nonzero and zero eigenspaces, we have obtained a full basis of $g$ ⁠, which is $(\cup_{j} {k_{j}, p_{j}}) \cup K_{z} \cup P_{z}$ ⁠. Refining once more, $span ((\cup_{j} {k_{j}}) \cup K_{z}) = k$ and $span ((\cup_{j} {p_{j}}) \cup P_{z}) = p$ ⁠.

E. The operators ${a d}_{H}, e^{{a d}_{H}}$ ⁠, and the subspaces $k, p$

In Sec. III D, we obtained the basis of $k$ and $p$ ⁠, in terms of ping pong matrices, by linearly combining eigenmatrices of the operator ad_H. We now illustrate the relationship of the basis of $k$ and $p$ under $e^{{a d}_{H}}$ ⁠, just like we illustrated the operator M_N in Sec. III A. In the k₁, …, k_N and p₁, …, p_N basis, we have the following:

e^{{a d}_{H}} [\begin{matrix} k_{1} \\ p_{1} \\ ⋮ \\ k_{N} \\ p_{N} \end{matrix}] = [\begin{matrix} \cosh α_{1} & \sinh α_{1} \\ \sinh α_{1} & \cosh α_{1} \\ ⋱ \\ \cosh α_{N} & \sinh α_{N} \\ \sinh α_{N} & \cosh α_{N} \end{matrix}] [\begin{matrix} k_{1} \\ p_{1} \\ ⋮ \\ k_{N} \\ p_{N} \end{matrix}] .

(3.8)

We are now ready to carefully investigate the map dΦ using (3.8).

Remark 3.4.

Results in Lie theory imply that the eigenmatrices x_j and θx_j of ad_H are independent of the choice of

H \in a

⁠. In other words, the complete basis of

g

and

k

⁠,

p

obtained above does not care about a specific choice of H. Furthermore, the eigenvalues ±α_j are functions of H, and these eigenvalue assigning functions

{\tilde{α}}_{j} : H \mapsto α_{j} \in R

are more properly called the restricted roots. It can be inferred from the separation of the basis that

k

⁠,

p

together form the whole tangent space

g

⁠,

g = k + p .

(3.9)

F. Symmetric spaces

The reader may have noticed that our discussions have focused on the Lie algebras rather than the Lie groups themselves. It is a point of fact that Lie groups are mostly useful to define the factorizations of our interest, but Lie algebras are where the Jacobian “lives,” and hence, this is the most important place to concentrate. For the interested reader, the subgroup K of G is picked such that its tangent space is exactly $k$ [one easy way to imagine such a subgroup is to define $K ≔ \exp (k)$ ], and we now obtain a symmetric space G/K.

It can be proven that for the noncompact Lie group, there exists a unique involution θ such that the subgroup K is the maximal compact subgroup of G. We call θ the Cartan involution, and (3.9) is called the Cartan decomposition. Furthermore, the subset $P ≔ \exp (p)$ plays an important role as its elements serve as representatives of the cosets in G/K. Regarding the identification of G/K as elements in P, refer to Remark 2.5, where we point out as an example, taking $G / K = GL (n, R) / O (n)$ that an element of G/K has the form of a coset gK, then gg^T may be a representative of the coset in $p$ ⁠. While some authors use ${(g g^{T})}^{1 / 2}$ ⁠, the key point being each choice is well defined independent of choice of representative.

G. When G is a compact Lie group

Upon considering the compact cases, it is helpful to make use of a certain duality between compact and noncompact symmetric spaces. We again start with a noncompact Lie group G and the Cartan involution θ. Let $g = k + p$ be the Cartan decomposition. Then, define a new space,

g_{C} ≔ k + i p,

(3.10)

where i is the imaginary unit. The result in Lie theory implies that the new vector space $g_{C}$ is the tangent space of a compact Lie group, say, G_C. In Table I, the first and third columns labeled $GL (n, R) / O (n)$ and O(p, q)/(O(p) × O(q)) are noncompact tangent spaces. Their compact duals are, respectively, the second and fourth columns labeled U(n)/O(n) and O(n)/(O(p) × O(q)).

Matrixwise, the ping pong matrices $k_{j} \in k, p_{j} \in p$ of $g$ are brought back to a new set of ping pong matrices $k_{j} \in k_{C}, i p_{j} \in p_{C}$ in $g_{C}$ ⁠. Let us denote them by ${\tilde{k}}_{j} ≔ k_{j}$ and ${\tilde{p}}_{j} ≔ i p_{j}$ ⁠. The role of the subspace $a$ is now played by $i a$ replacing ad_H by ad_iH. We deduce a set of similar relationships for ${\tilde{k}}_{j}, {\tilde{p}}_{j}$ under ad_iH,

{a d}_{i H} ({\tilde{k}}_{j}) = α_{j} {\tilde{p}}_{j}, {a d}_{i H} ({\tilde{p}}_{j}) = - α_{j} {\tilde{k}}_{j} .

In matrix form,

{a d}_{i H} [\begin{matrix} {\tilde{k}}_{j} \\ {\tilde{p}}_{j} \end{matrix}] = [\begin{matrix} 0 & α_{j} \\ - α_{j} & 0 \end{matrix}] [\begin{matrix} {\tilde{k}}_{j} \\ {\tilde{p}}_{j} \end{matrix}],

(3.11)

which leads to the compact version of (3.6) and (3.7),

\exp ({a d}_{i H}) [\begin{matrix} {\tilde{k}}_{j} \\ {\tilde{p}}_{j} \end{matrix}] = [\begin{matrix} \cos α_{j} & \sin α_{j} \\ - \sin α_{j} & \cos α_{j} \end{matrix}] [\begin{matrix} {\tilde{k}}_{j} \\ {\tilde{p}}_{j} \end{matrix}] .

(3.12)

At the group level, the symmetric spaces G/K and G_C/K are called the duals of each other, and they appear in the same row of standard symmetric space charts. An example of eigenmatrices x_j, θx_j and ping pong matrices for some symmetric spaces and their duals are presented in Table I.

H. Jacobian of the map Φ

We provide a generalized algorithm for finding a Jacobian of the decomposition Φ(Q, Λ) = QΛQ⁻¹ [as we defined in (2.5)], where $Λ \in A ≔ \exp (a), Q \in K$ ⁠. $k$ and $p$ from Sec. III D are the tangent spaces of K and P, respectively. As mentioned, we follow Helgason’s derivation (Ref. 28, Theorem 5.8 of Chap. I) and start by directly translating his proof into simple linear algebra terms. In Table II, we have Helgason’s derivation (left) compared in the same row with linear algebra (Right). Table II is using the noncompact symmetric space G/K but the compact case is identical with replacing sin α_j by sinh α_j.

TABLE II.

Line-by-line translation of the classical proof to linear algebra proof.

Classical notation (Ref. 28, p. 187, Proof of Theorem 5.8, Chap. I)

Linear algebra notation (matrix factorizations)

Definitions

Φ: K × A → G/K

\tilde{Φ}

⁠: K × A → P

Φ: (k, a) ↦ kaK

\tilde{Φ}

⁠: (Q, Λ) ↦ QΛQ⁻¹ (⁠

Λ^{\frac{1}{2}} = a

⁠, Q = k)

d τ (g_{0}) : {(G / K)}_{o} \to {(G / K)}_{g_{0} \cdot o}

d \tilde{τ} (g_{0}) : X \mapsto g_{0} X {(θ g_{0})}^{- 1}

d π : g \to {(G / K)}_{o}

(θk = k, k ∈ K, θp = p⁻¹, p ∈ P)

At k ∈ K, fix a tangent vector

d τ (k) T_{i}^{α}

At Q ∈ K, fix a tangent vector dQ

At Id, basis element

T_{i}^{α} \in k

At Id, basis element

Q^{- 1} d Q = k_{j} \in k

Derivations

2 d Φ (d τ (k) T_{i}^{α}, 0)

^a

d \tilde{Φ} (d Q, 0) = d (Q Λ Q^{- 1})

(with dΛ = 0)

= d π (2 k T_{i}^{α} a)

= dQΛQ⁻¹ + QΛdQ⁻¹

= d τ (k a) d π (2 Ad (a^{- 1}) T_{i}^{α})

^b

= d \tilde{τ} (Q Λ^{\frac{1}{2}}) [Λ^{- \frac{1}{2}} (Q^{- 1} d Q Λ + Λ d Q^{- 1} Q) Λ^{- \frac{1}{2}}]

^c

= d τ (k a) d π (Ad (a^{- 1}) T_{i}^{α} - Ad (a) T_{i}^{α})

= d \tilde{τ} (Q Λ^{\frac{1}{2}}) [Λ^{- \frac{1}{2}} k_{j} Λ^{\frac{1}{2}} - Λ^{\frac{1}{2}} k_{j} Λ^{- \frac{1}{2}}]

[Let H be such that

\exp (H) = a = Λ^{\frac{1}{2}}

]

[Note that

d \tilde{τ} (Q Λ^{\frac{1}{2}}) X = Q Λ^{\frac{1}{2}} X Λ^{\frac{1}{2}} Q^{- 1}

]

= d τ (k a) d π (e^{- ad H} T_{i}^{α} - e^{ad H} T_{i}^{α})

= d \tilde{τ} (Q Λ^{\frac{1}{2}}) [\exp (H^{T} \otimes I - I \otimes H) k_{j})

(- \exp (I \otimes H - H^{T} \otimes I) k_{j}]

[by ()]

= d τ (k a) d π (- α {(H)}^{- 1} [H, T_{i}^{α}] 2 \sinh α (H))

= d \tilde{τ} (Q Λ^{\frac{1}{2}}) [(- 2 \sinh α_{j}) p_{j}]

[by ()]

^a

Since $Λ^{\frac{1}{2}} = a$ ⁠, we have $2 d Φ = d \tilde{Φ}$ ⁠.

^b

This is $(d τ (k a) ◦ d π) (Ad (a^{- 1}) T_{i}^{α})$ ⁠.

^c

Both dQΛQ⁻¹ and QΛdQ⁻¹ are at QΛQ⁻¹ and should be brought back to identity (inside bracket).

From the last line of Table II, we can finish the story with two different directions, depending on the choice of the volume measure. First, if we use a G-invariant measure (the “canonical measure”) of P, the measure is invariant under the map dτ or $d \tilde{τ}$ (by definition of the invariant measure). Thus, we can disregard $d \tilde{τ} (Q Λ^{\frac{1}{2}})$ [or dτ(ka)] so that the Jacobian of $d \tilde{Φ}$ (or dΦ) only depends on the differential map k_j ↦ (sinh α_j)p_j. Since ∪_j{k_j} and ∪_j{p_j} are both orthonormal bases, we obtain the Jacobian (2.2),

\prod_{α \in Σ^{+}} \sinh α (H) .

Note that eigenvalues ±α_j belong to x_j and θx_j have the same corresponding k_j. [see (3.4) and above.] Thus, we only take the positive roots Σ⁺ above.

The second choice of measure is the Euclidean measure, which is a wedge product of independent entrywise differentials. In this case, the procedure is identical up to the factor sinh α_j, but the map $d \tilde{τ} (Q Λ^{\frac{1}{2}})$ [equivalently dτ(ka)] cannot be ignored. One needs to carefully compute the differential map $d \tilde{τ} (Q Λ^{\frac{1}{2}}) p_{j} = Q Λ^{\frac{1}{2}} p_{j} Λ^{\frac{1}{2}} Q^{- 1}$ under the Euclidean measure. We can further use the fact that conjugation by the matrix Q always preserves the Euclidean measure, since the subgroup K is always a set of matrices with an orthogonal/unitary type of property. Thus, one needs to compute the map $p_{j} \mapsto Λ^{\frac{1}{2}} p_{j} Λ^{\frac{1}{2}}$ and multiply its Jacobian by $\prod_{α \in Σ^{+}} \sinh α (H)$ ⁠.

Remark 3.5.

For the compact Lie group G, we have sinh α_j replaced by sin α_j everywhere. Moreover, the last Jacobian computation step $p_{j} \mapsto Λ^{\frac{1}{2}} p_{j} Λ^{\frac{1}{2}}$ can be omitted for the compact cases, since $Λ^{\frac{1}{2}}$ is an orthogonal/unitary matrix for the compact cases. The map $d \tilde{τ} (Λ^{\frac{1}{2}})$ preserves the Euclidean measure as $d \tilde{τ} (Q)$ ⁠.

I. Extension to the generalized Cartan decomposition

In the previous paragraphs, we studied the Jacobian of the usual Cartan decomposition. We now proceed to consider the generalized Cartan decomposition (Theorems 2.1 and 2.2), its Jacobian (2.2), (2.3), and the extension of Table II. The derivations are analogous, analyzing subspaces of $g$ ⁠, but one should now proceed with four tangent subspaces, $k_{τ} \cap k_{σ}$ ⁠, $k_{τ} \cap p_{σ}$ ⁠, $p_{τ} \cap k_{σ}$ ⁠, and $p_{τ} \cap p_{σ}$ ⁠. Earlier work on these Jacobian related derivations may be found in Refs. 23 and 44. The maximal subspace $a$ is now defined inside $p_{τ} \cap p_{σ}$ ⁠. We start with the same strategy: the tangent space $g$ is decomposed into the eigenspaces of the linear operator ad_H with $H \in a$ ⁠. The eigenvalues ±α_j still come in pairs, but we have two eigenmatrices x_j, τσx_j for eigenvalue α_j and two eigenmatrices τx_j, σx_j for eigenvalue −α_j. We define four vectors v₁, v₂, w₁, w₂ with the same roles as k_j and p_j played before,

\begin{matrix} v_{1} ≔ x_{j} + τ x_{j} + σ x_{j} + τ σ x_{j} \in k_{τ} \cap k_{σ}, v_{2} ≔ x_{j} - τ x_{j} - σ x_{j} + τ σ x_{j} \in p_{τ} \cap p_{σ}, \\ w_{1} ≔ x_{j} - τ x_{j} + σ x_{j} - τ σ x_{j} \in p_{τ} \cap k_{σ}, w_{2} ≔ x_{j} + τ x_{j} - σ x_{j} - τ σ x_{j} \in k_{τ} \cap p_{σ}, \end{matrix}

and these have similar ping pong relationships by ad_H like k_j and p_j,

\begin{matrix} {a d}_{H} (v_{1}) = α_{j} v_{2}, {a d}_{H} (v_{2}) = α_{j} v_{1}, \\ {a d}_{H} (w_{1}) = α_{j} w_{2}, {a d}_{H} (w_{2}) = α_{j} w_{1} . \end{matrix}

We can similarly extend (3.8) and other relationships and proceed as in Table II to obtain (2.2) and (2.3).

IV. RANDOM MATRIX ENSEMBLES: COMPACT AND NONCOMPACT

A. Compact symmetric spaces

In compact cases, the random matrices could be simply determined from the Haar measure of the compact Lie group G,^12,13 since the compactness of G turns the Haar measure into a probability measure. In Secs. V, VI, VII, we discuss random matrix ensembles based on ten types of Riemannian symmetric space classification by Cartan. For the triple (G, K_σ, K_τ), we start with the cases where G/K_σ and G/K_τ are of the same types in Secs. V and VI. Then, in Sec. VII, we will discuss the “mixed types” where G/K_σ and G/K_τ are different types under Cartan’s classification.

B. Noncompact symmetric spaces

Sections VIII and IX discuss classical random matrix ensembles associated with noncompact symmetric spaces. Hermite and Laguerre eigenvalue joint densities arise as result of (2.2) using Theorem 2.4 on noncompact symmetric spaces. As opposed to compact Lie groups and symmetric spaces where the Haar measure or G-invariant measure can be normalized by a constant to a probability measure, invariant measures on noncompact manifolds cannot be normalized to one by constants. A normalizing factor $S$ should be introduced to complete the construction of a probability measure. Therefore, random matrices on a noncompact manifold face an innate problem if we proceed analogous to Secs. V and VI:

The choice of the probability measure on noncompact G/K is not unique.

In Ref. 13, Dueñez also addressed this problem along the noncompact duals.

As we push the measure forward to the subgroup A, the resulting measure should be a symmetric function of independent generators of A. Hence, the probability measure $I (g)$ of the random matrix ensemble is the Haar or G-invariant measure on G or G/K, multiplied by some symmetric function $S$ on A,

I (g) = S (a) μ (g),

where g = k₁ak₂ or g = kak⁻¹ and μ(g) is an invariant measure. Using (2.2), the measure on A is induced,

I (g) = d k \cdot S (a) (\prod_{α \in Σ^{+}} \sinh α (H)) d H_{1} \dots d H_{dim (A)},

which means that even though the measure $I$ changes, the measure on A still differs only by a normalization function. The traditional choice of $S$ has been made such that $I (g)$ can be constructed from independent Gaussian distributions endowed on matrix entries. In fact, one could also endow a Gaussian distribution on the Riemannian manifold (symmetric space) itself.⁶⁵

An alternative approach that appears in Ref. 6 is to put a probability measure on the tangent space of the symmetric space, $p$ ⁠. In particular, independent Gaussian distribution endowed on the elements of $p$ give rise to Hermite and Laguerre ensembles by Theorem 2.7. We will follow this alternative approach.

C. Non-probability measure of noncompact groups

As discussed in Sec. IV B, the Haar measure of a noncompact group G or a noncompact symmetric space G/K is not a probability measure. However, we can force an analog of a random matrix theory. Imagine, for example, a noncompact K₁AK₂ decomposition G = K_σAK_τ with $(G, K_{σ}, K_{τ}) = (GL (n, R), O (n), O (p, q))$ ⁠. This is called the hyperbolic SVD⁶⁶ where any real invertible matrix M is factored into the product of an orthogonal matrix O, a positive diagonal matrix Λ, and an indefinite orthogonal matrix V. From the Haar measure and (2.2) of $GL (n, R)$ ⁠, one obtains the Jacobian,

\prod_{\begin{array}{c} 1 \leq j < k \leq p \\ p < j < k \leq n \end{array}} | λ_{j} - λ_{k} | \prod_{\begin{array}{c} 1 \leq j \leq p \\ p < j \leq n \end{array}} | λ_{j} + λ_{k} | \prod_{j = 1}^{n} | λ_{j} |^{- \frac{2 n + 1}{2}} d λ_{1} \dots d λ_{n},

where λ_j is the squared diagonal entries of Λ for all j’s.

One can impose a Gaussian-like density function (although not a probability density) on the group $GL (n, R)$ ⁠, such as exp(−tr(gI_p,qg^T)/2)∏dg_jk, where I_p,q = diag(I_p, −I_q). In terms of independent entries of g, this is

\prod_{first p columns} e^{- g_{j k}^{2} / 2} \prod_{last q columns} e^{g_{j k}^{2} / 2} \prod d g_{j k} .

(4.1)

Since the Haar measure of $GL (n, R)$ is |det(g)|⁻ⁿ ∏dg_jk, (4.1) becomes [after integrating out O(n) and O(p, q)]

\prod_{j < k} | λ_{j} - λ_{k} | \prod_{j = 1}^{n} | λ |^{- \frac{n + 1}{2}} e^{- \sum λ_{j} / 2} d λ_{1} \dots d λ_{n},

where λ₁, …, λ_p ≥ 0 are the first p squared diagonal values of Λ and λ_p+1, …, λ_n ≤ 0 are the last q squared diagonal values of Λ, multiplied by −1. Extending this approach to find a proper random matrix probability measure on noncompact Lie groups and symmetric spaces with joint probability densities on the subgroup A is still an open problem.

V. COMPACT AI, A, and AII: CIRCULAR ENSEMBLES

The joint probability density of the circular ensemble is (β = 1, 2, 4)

E_{n}^{(β)} (θ) \propto \prod_{j < k} | e^{i θ_{j}} - e^{i θ_{k}} |^{β} .

Circular ensembles β = 1, 2, and 4 (COE, CUE, and CSE) arise as the eigenvalues of special unitary matrices. As we discuss in the Introduction, circular ensembles are completely classified by (compact) symmetric spaces of the types AI, A, and AII, respectively.^5,13 The K₁AK₂ decomposition associated with each symmetric space recovers the KAK decomposition. The restricted root system (and dimensions) of AI, A, and AII are given as the following (1 ≤ j < k ≤ n):

(5.1)

Since we have compact symmetric spaces, we use (2.3) from either Theorem 2.2 or 2.4 with these root systems.

A. Compact AI, $β = 1$ COE

The compact symmetric space AI is G/K = U(n)/O(n). The involution on U(n) has no free parameter and the K₁AK₂ decomposition is equivalent to the KAK decomposition of U(n)/O(n). (In other words, we only have Cartan’s coordinate system.) The maximal Abelian torus A is

A = {Diagonal matrices with entries e^{i h_{j}}, where h_{j} \in R} .

From the KAK decomposition, we obtain U = O₁DO₂, a factorization of a unitary matrix U into the product of two orthogonal matrices O₁, O₂ ∈ O(n) and a unit complex diagonal matrix D ∈ A. This decomposition first appears in Ref. 67, and we will call this the ODO decomposition. The corresponding Jacobian (up to constant) from (2.3) using (5.1), β = 1 is (with the change of variables θ_j = 2h_j)

(\prod_{j < k} \sin (h_{j} - h_{k})) d h_{1} \dots d h_{n} \propto \prod_{j < k} | e^{i θ_{j}} - e^{i θ_{k}} | d θ_{1} \dots d θ_{n} .

This is the joint density of the COE. In other words, doubled angles in the diagonal of D from the ODO decomposition of a Haar distributed unitary matrix is the COE distribution. Moreover, if we identify G/K as the set of unitary symmetric matrices P, the map (2.5) is the factorization S = OΛO^T, the eigendecomposition of a unitary symmetric matrix S with real eigenvectors O. In terms of Remark 2.5, U = O₁DO₂ becomes $S = U U^{T} = O_{1} D^{2} O_{1}^{T}$ ⁠, where Λ = D². To obtain the COE, we can utilize both factorizations:

Two times the angles of the unit diagonal values of D from the ODO decomposition of U ∈ Haar(U(n)).
The angles of the (unit) eigenvalues of a unitary symmetric matrix obtained from UU^T, U ∈ Haar(U(n)).

Remark 5.1.

The second algorithm above would be obvious since the days of Dyson,^1,4 while we are not aware of the first algorithm appearing in the literature.

B. Compact A, $β = 2$ CUE

The symmetric space of compact type A is G/K = U(n) × U(n)/U(n). The restricted root system returns to the usual root system A_n of the classical semisimple Lie algebra. A maximal torus of U(n) is a Cartan subalgebra of U(n). Weyl’s integration formula agrees with (2.3) obtaining the CUE, which is the eigenvalues of a Haar distributed unitary matrix. The derivation of the CUE can be found in many random matrix textbooks.^15,64,68

C. Compact AII, $β = 4$ CSE

The involution $X \mapsto - J_{n}^{T} X^{T} J_{n}$ ⁠, where $J_{n} ≔ [\begin{matrix} 0 & I_{n} \\ - I_{n} & 0 \end{matrix}]$ on the tangent space of U(2n), results in the symmetric space U(2n)/Sp(n), where $Sp (n) = Sp (2 n, C) \cap U (2 n)$ ⁠. A choice of maximal Abelian torus A is

A = {diag (\tilde{D}, \tilde{D}) : \tilde{D} = diag (e^{i h_{1}}, \dots, e^{i h_{n}}), h_{j} \in R} .

Again from the KAK decomposition, we obtain U = Q₁DQ₂, a factorization of a 2n × 2n unitary matrix U into the product of two unitary symplectic matrices Q₁, Q₂ ∈ Sp(n) and a unit complex diagonal matrix D ∈ A. We call this the QDQ decomposition. The corresponding Jacobian from (2.3) using (5.1) (β = 4) is

(\prod_{j < k} \sin^{4} (h_{j} - h_{k})) d h_{1} \dots d h_{n} \propto \prod_{j < k} | e^{i θ_{j}} - e^{i θ_{k}} |^{4} d θ_{1} \dots d θ_{n},

with the change of variables θ_j = 2h_j. This is the CSE distribution. Similarly, as in Sec. V A, the eigendecomposition of the unitary skew-Hamiltonian matrix obtained by $U J_{n} U^{T} J_{n}^{T}$ ⁠, U ∈ Haar(2n) is equivalent to the map (2.5). Two numerical algorithms for sampling the CSE are as follows:

Two times the angles of the first n unit diagonal values of D from the QDQ decomposition of U ∈ Haar(U(2n)).
The angles of the first n (unit) eigenvalues of a unitary skew-Hamiltonian matrix obtained by $U J_{n} U^{T} J_{n}^{T}$ with U ∈ Haar(U(2n)).

VI. COMPACT BDI, AIII, and CII: JACOBI ENSEMBLES

The joint probability density of the Jacobi ensemble is (β = 1, 2, 4),

J_{α_{1}, α_{2}}^{(β), m} (x) \propto \prod_{j < k} | x_{j} - x_{k} |^{β} \prod_{j = 1}^{m} x_{j}^{α_{1}} {(1 - x_{j})}^{α_{2}} .

In Refs. 12 and 13, Jacobi ensembles β = 1, 2, 4 arise from the KAK decompositions of seven compact symmetric spaces, BDI, AIII, CII, DIII, BD, C, and CI. In particular, types BDI, AIII, and CII give multiple Jacobi densities as follows (for integers p ≥ q):

\prod_{j < k} | x_{j} - x_{k} |^{β} \prod_{j = 1}^{q} x_{j}^{\frac{β}{2} - 1} {(1 - x_{j})}^{\frac{β (p - q + 1)}{2} - 1},

and the powers of x_j’s are fixed to $\frac{β}{2} - 1$ ⁠. The remaining four cases add four more parameter points, which could be found in Refs. 12 and 13. In this paper, we omit these four cases as these do not have any further results, as they only have Cartan’s coordinates (no free parameter for the Cartan involution).

The K₁AK₂ decomposition G = K_τAK_σ of the compact types BDI-I, AIII-III, CII-II are exactly the CS decomposition (CSD)^69,70 of orthogonal, unitary, and unitary symplectic matrices, respectively. The decomposition Φ of the symmetric space (Theorem 2.4) is the GSVD coordinate systems we discussed in Secs. I B and II C. Assume r ≥ p ≥ q ≥ s and n = p + q = r + s throughout this section. We note that with the KAK decomposition, only the cases p = r, q = s are obtained for the CSD. The root system associated with the K₁AK₂ decomposition is the following (1 ≤ j < k ≤ s):

(6.1)

For all three β, we have the identical maximal Abelian subgroup A,

A = {n \times n matrices with the block structure [\begin{matrix} C & S \\ I_{p - q} \\ - S & C \end{matrix}]},

where $C, S \in R^{s \times s}$ are diagonal matrices with cosine, sine values of θ₁, …, θ_s on diagonal entries, respectively.

A. Compact BDI-I, $β = 1$ Jacobi

With the involution X ↦ I_p,qXI_p,q on the tangent space of O(n), we obtain the symmetric space BDI, G/K = O(n)/(O(p) × O(q)), where I_p,q ≔ diag(I_p − I_q). With two symmetric pairs [O(n), O(p) × O(q)] and [O(n), O(r) × O(s)], we obtain the K₁AK₂ decomposition BDI-I,

[\begin{matrix} n - by - n \\ Orthogonal \end{matrix}] = [\begin{matrix} O_{p} \\ O_{q} \end{matrix}] [\begin{matrix} C & S \\ I_{n - 2 s} \\ - S & C \end{matrix}] [\begin{matrix} O_{r} \\ O_{s} \end{matrix}] .

This is the real CSD. [Equivalently, one can imagine the GSVD of (1.2).] From (2.3) using (6.1) β = 1, we obtain the Jacobian

d μ (H) \propto \prod_{j < k} (\sin (θ_{j} - θ_{k}) \sin (θ_{j} + θ_{k})) \prod_{j} ({(\sin θ_{j})}^{(p - s)} {(\cos θ_{j})}^{(q - s)}) d θ_{1} \dots d θ_{s} .

Using trigonometric identities with change of variables $x_{j} = \cos^{2} θ_{j} = \frac{1 + \cos (2 θ_{j})}{2}$ ⁠,

d μ (H) \propto \prod_{j < k} | x_{j} - x_{k} | \prod_{j = 1}^{s} x_{j}^{\frac{1}{2} (q - s + 1) - 1} {(1 - x_{j})}^{\frac{1}{2} (p - s + 1) - 1} d x_{1} \dots d x_{s},

which is the joint density of the β = 1 Jacobi ensemble $J_{α_{1}, α_{2}}^{(1), s}$ if we let $α_{1} = \frac{1}{2} (q - s + 1) - 1$ ⁠, $α_{2} = \frac{1}{2} (p - s + 1) - 1$ ⁠. This result agrees with Ref. 20, Theorem 1.5, where the squared CSD cosine values of a Haar distributed orthogonal matrix are distributed as β = 1 Jacobi ensemble. Moreover, recall the fact that the QL decomposition G = QL (a lower triangular analog of the QR decomposition) of an n × n independent Gaussian matrix G obtains a Haar distributed orthogonal matrix Q. Since the GSVD^18,19 is equivalent to the combination of the QL decomposition and the CSD, one can take the GSVD of a real independent Gaussian matrix to obtain the same β = 1 Jacobi ensemble. Two associated numerical algorithms are as follows (a = q − s, b = p − s):

The squared CSD cosine values of a Haar distributed m × m orthogonal matrix (m = 2s + a + b) with row/column partitions (s + a, s + b) and (s, s + a + b).
The squared cosine values, where the tangent values are the generalized singular values of real (s + a) × s and (s + b) × s Gaussian matrices.

B. Compact AIII-III, β = 2 Jacobi

Two symmetric pairs of compact AIII type are [U(n), U(p) × U(q)] and [U(n), U(r) × U(s)]. The K₁AK₂ decomposition of the group G is the CSD of unitary matrices and the decomposition of G/K_σ = U(n)/(U(r) × U(s)) are the complex GSVD described in Sec. I B and Eq. (1.2). Using (2.3) with the root system (6.1), β = 2, and change of variables x_j = cos² θ_j as above, we obtain the Jacobian

\begin{array}{l} \prod_{j < k} {(\sin (θ_{j} - θ_{k}) \sin (θ_{j} + θ_{k}))}^{2} \prod_{j} ({(\sin θ_{j})}^{2 (p - s)} {(\cos θ_{j})}^{2 (q - s)} \sin (2 θ_{j})) d θ_{1} \dots d θ_{s} \\ \propto \prod_{j < k} | x_{j} - x_{k} |^{2} \prod_{j} x_{j}^{q - s} {(1 - x_{j})}^{p - s} d x_{1} \dots d x_{s}, \end{array}

which is the β = 2 Jacobi density $J_{α_{1}, α_{2}}^{(2), s}$ with α₁ = q − s, α₂ = p − s. Numerically, the following could be utilized to obtain β = 2 Jacobi densities (a = q − s, b = p − s):

The squared CSD cosine values of a Haar distributed m × m unitary matrix (m = 2s + a + b) with row/column partitions (s + a, s + b) and (s, s + a + b).
The squared cosine values, where the tangent values are the generalized singular values of complex (s + a) × s and (s + b) × s Gaussian matrices.

C. Compact CII-II, β = 4 Jacobi

Jacobi densities with β = 4 are similarly obtained from two symmetric spaces Sp(n)/(Sp(p) × Sp(q)) and Sp(n)/(Sp(r) × Sp(s)), where both are compact type CII. We identify Sp(n) as the quaternionic unitary group, $U (n, H) ≔ {g \in GL (n, H) | g^{D} g = I_{n}}$ ⁠. The K₁AK₂ decomposition is the CSD of a quaternionic unitary matrix. Using (2.3) with the root system (6.1) β = 4, we obtain the following Jacobian with the change of variables x_j = cos² θ_j:

\begin{array}{l} \prod_{j < k} {(\sin (θ_{j} - θ_{k}) \sin (θ_{j} + θ_{k}))}^{4} \prod_{j} ({(\sin θ_{j})}^{4 (p - s)} {(\cos θ_{j})}^{4 (q - s)} \sin^{3} (2 θ_{j})) d θ_{1} \dots d θ_{s} \\ \propto \prod_{j < k} | x_{j} - x_{k} |^{4} \prod_{j} x_{j}^{2 (q - s) + 1} {(1 - x_{j})}^{2 (p - s) + 1} d x_{1} \dots d x_{s}, \end{array}

which is the β = 4 Jacobi density $J_{α_{1}, α_{2}}^{(4), s}$ with α₁ = 2(q − s) + 1, α₂ = 2(p − s) + 1. The associated numerical algorithm is the following (a = q − s, b = p − s):

The squared cosine CS values of a Haar distributed m × m quaternionic unitary matrix (m = 2s + a + b) with row/column partitions (s + a, s + b) and (s, s + a + b).

Remark 6.1.

Again, one can use the GSVD on quaternionic Gaussian matrices to obtain the classical β = 4 Jacobi ensemble.

VII. COMPACT MIXED TYPES: MORE CIRCULAR AND JACOBI

In this section, we show even more cases such that a single symmetric space leading to multiple random matrix theories. We introduce K₁AK₂ decompositions with two compact symmetric spaces, each from different Cartan types. The classification of such K₁AK₂ decompositions is studied in Ref. 47, with the computation of corresponding root systems. As always the names of these decompositions are combinations of two Cartan types, i.e., AI-II represents (G, K_σ, K_τ) = (U(2n), O(2n), Sp(2n)).

A. Compact AI-II

The two compact symmetric spaces are types AI and AII, U(2n)/O(2n) and U(2n)/Usp(2n). A maximal Abelian subalgebra $a \subset p_{σ} \cap p_{τ}$ is the set of all matrices diag(iθ₁, …, iθ_n, iθ₁, …, iθ_n) for $(θ_{1}, \dots, θ_{n}) \in R^{n}$ ⁠. The subgroup A is the following:

A = {diag (\tilde{D}, \tilde{D}) : \tilde{D} = diag (e^{i θ_{1}}, \dots, e^{i θ_{n}})} .

The root system is given as

(7.1)

Using (2.3), we obtain the Jacobian (ξ_j = 4θ_j),

| e^{i ξ_{j}} - e^{i ξ_{k}} |^{2} d ξ_{1} \dots d ξ_{n},

which is the joint probability density of the CUE. Hence, we obtain another sampling method for the CUE.

B. Compact AI-III, CI-II

The two symmetric spaces in each case are the following:

\begin{array}{l} G / K_{τ}, G / K_{σ} & = U (n) / O (n), U (n) / (U (p) \times U (q)), \\ G / K_{τ}, G / K_{σ} & = U (n, H) / U (n), U (n, H) / (U (p, H) \times U (q, H)) . \end{array}

The subgroup A is computed as follows:

A = {n \times n matrices with the block structure [\begin{matrix} C & η S \\ I_{p - q} \\ η S & C \end{matrix}]},

where C, S are q × q diagonal matrices with cosine and sine values of q angles θ₁, …, θ_q on their diagonals. The imaginary unit η is i for AI-III (β = 1) and η = j, k for CI-II (β = 2). [If we select the subgroup K of $U (n, H) / U (n)$ to be the unitary group with the imaginary unit j, we could also obtain η = i.] The root system is the following (β = 1, 2):

(7.2)

Using (2.3) with the above root system above, we obtain the following Jacobian:

\prod_{j < k} | x_{j} - x_{k} |^{β} \prod_{j = 1}^{q} x_{j}^{\frac{β (p - q + 1)}{2} - 1} {(1 - x_{j})}^{\frac{β - 1}{2}},

(7.3)

where x_j = sin² 2θ_j for all j. The β = 1 case of (7.3) can be obtained from the CS decomposition approach too, with (n + 1) × (n + 1) orthogonal matrix and partitions (p, q + 1) and (p + 1, q) [see Fig. 4]. The parameters of β = 2 (7.3) cannot be obtained by the complex CSD and, thus, fall outside of the classical parameters.

C. Compact DI-III, AII-III

Another family of the K₁AK₂ decomposition arise from the following pairs of compact symmetric spaces (β = 2, 4):

\begin{array}{l} G / K_{τ}, G / K_{σ} & = O (2 n) / U (n), O (2 n) / (O (2 p) \times O (2 q)), \\ G / K_{τ}, G / K_{σ} & = U (2 n) / U (n, H), U (2 n) / (U (2 p) \times U (2 q)) . \end{array}

Under Cartan’s classification, they are types DI-III and AII-III, respectively. The subgroup A can be computed as

A = {2 n \times 2 n matrices with the block structure [\begin{matrix} I_{p - q} \\ C \otimes I_{2} & S \otimes J_{1} \\ I_{p - q} \\ S \otimes J_{1} & C \otimes I_{2} \end{matrix}]},

where I₂ is the 2 × 2 identity matrix, $J_{1} = [\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}]$ ⁠, and C, S are q × q diagonal matrices with cosines and sines of θ₁, …, θ_q on their diagonals. The root system is given as follows (β = 2, 4):

(7.4)

Again, using (2.3) with the root system above, we obtain the following Jacobian, with the change of variables x_j = sin² θ_j for all j:

\prod_{j = 1}^{q} x_{j}^{\frac{β (p - q + 2)}{4} - 1} {(1 - x_{j})}^{\frac{β - 4}{4}} \prod_{j < k} | x_{j} - x_{k} |^{β} .

(7.5)

They are β = 2, 4 Jacobi ensembles. Both cases could not be obtained from the classical CSD approach, so they are all non-classical parameters of the Jacobi ensemble. To see this at once, we compare three β = 2 Jacobi densities each from Secs. VI B, VII B and VII C. Figure 6 shows the possible parameters α₁, α₂ of the β = 2 Jacobi ensemble obtained from each approach.

FIG. 6.

View large Download slide

The parameter space (α₁, α₂) ∈ (−1,∞)² of the β = 2 Jacobi ensemble covered by symmetric spaces. The GSVD coordinate systems on the complex Grassmannian manifold (AIII-III) discussed in Sec. VI covers red dots. A new coordinate system on the quaternionic (respectively, real) Grassmannian manifold discussed in Sec. VII B (respectively, Sec. VII C) of type CI-II (respectively, DI-III) represent blue (respectively, green) dots.

VIII. NONCOMPACT AI, A, AND AII: HERMITE ENSEMBLES

While Sec. VII contains essentially new random matrix theories, Secs. VIII and IX review the Hermite and Laguerre ensembles for completeness.^6–9,59

The joint probability density of the Hermite ensemble is (β = 1, 2, 4),

H_{n}^{(β)} (λ) \propto \prod_{j < k} | λ_{j} - λ_{k} |^{β} \prod_{j = 1}^{n} e^{- λ_{j}^{2} / 2} .

Hermite ensembles β = 1, 2, and 4 (GOE, GUE, and GSE) arise as the eigenvalues of symmetric, Hermitian, and self-dual Gaussian matrices. Hermite ensembles can be thought as the Gaussian measure endowed on the tangent space of noncompact symmetric spaces of the types AI, A, and AII. The connection between these symmetric spaces and Hermite ensembles are made by Theorem 2.7. The decomposition Ψ (2.6) in Theorem 2.7 is the eigendecomposition of symmetric, Hermitian, and self-dual matrices. The maximal Abelian subalgebra $a$ is the collection of all real diagonal matrices, diag(h₁, …, h_n). The restricted root system is the following (1 ≤ j < k ≤ n):

(8.1)

A. Noncompact AI, β = 1 GOE

The dual of the compact symmetric space type AI, the noncompact symmetric space type AI, is $G / K = GL (n, R) / O (n)$ ⁠, represented by the set $S_{n}$ of all symmetric positive definite matrices. The tangent space at the identity of $S_{n}$ ⁠, $p$ ⁠, is the set of all real symmetric matrices. The Gaussian measure on $p$ is, for $p \in p$ ⁠, exp(−tr(p^Tp)/2)dp ,where dp is the Euclidean measure on $p$ ⁠. From (2.7) using (8.1) β = 1, we obtain (integrate out dk)

\exp (- t r (p^{T} p) / 2) d p \propto \prod_{j < k} | λ_{j} - λ_{k} | \prod_{j = 1}^{n} e^{- λ_{j}^{2} / 2} d λ_{1} \dots d λ_{n}

for the eigenvalues of p, λ_j = h_j. This is the joint density of the GOE.

B. Noncompact A, β = 2 GUE

The noncompact symmetric space type A is $G / K = GL (n, C) / U (n)$ ⁠, represented by $H_{n}$ ⁠, the set of all Hermitian positive definite matrices. The tangent space at the identity of $H_{n}$ ⁠, $p$ ⁠, is the set of all complex Hermitian matrices. The Gaussian measure on $p$ is, for $p \in p$ ⁠, exp(−tr(p^Hp)/2)dp, where dp is the (real) Euclidean measure on $p$ ⁠. From (2.7) using (8.1) β = 2, we obtain

\exp (- t r (p^{H} p) / 2) d p \propto \prod_{j < k} | λ_{j} - λ_{k} |^{2} \prod_{j = 1}^{n} e^{- λ_{j}^{2} / 2} d λ_{1} \dots d λ_{n}

for the eigenvalues of p, λ_j = h_j. This is the joint density of the GUE.

C. Noncompact AII, β = 4 GSE

The noncompact symmetric space type AII is $G / K = GL (n, H) / U (n, H)$ ⁠. We use $U (n, H)$ instead of Sp(n) to clearly indicate the quaternionic realization. G/K can be represented by the set of all quaternionic self-dual positive definite matrices, ${QH}_{n}$ ⁠. Again, the tangent space at the identity $p$ is the set of all quaternionic self-dual matrices. The Gaussian measure on $p$ is, for $p \in p$ ⁠, exp(−tr(p^Dp)/2)dp, where dp is the (real) Euclidean measure on $p$ ⁠. From (2.7) using (8.1) β = 4, we obtain

\exp (- t r (p^{D} p) / 2) d p \propto \prod_{j < k} | λ_{j} - λ_{k} |^{4} \prod_{j = 1}^{n} e^{- λ_{j}^{2} / 2} d λ_{1} \dots d λ_{n}

for the eigenvalues of p, λ_j = h_j. This is the joint density of the GSE.

IX. NONCOMPACT BDI, AIII, and CII: LAGUERRE ENSEMBLES

The joint probability density of the Laguerre ensemble is (β = 1, 2, 4)

L_{α, m}^{(β)} (λ) \propto \prod_{j < k} | λ_{j} - λ_{k} |^{β} \prod_{j = 1}^{m} λ_{j}^{α} e^{- λ_{j} / 2} .

Laguerre ensembles β = 1, 2, 4 arise from Theorem 2.7 applied to noncompact symmetric spaces BDI, AIII, CII, DIII, BD, C, and CI. The last four cases of types DIII, BD, C, and CI are well-studied in Ref. 6, and we again omit these cases as discussed in Sec. VI. In particular, the first three symmetric spaces give the following Laguerre densities (β = 1, 2, 4 and p ≥ q):

\prod_{j < k} | λ_{j} - λ_{k} |^{β} \prod_{j = 1}^{q} λ_{j}^{\frac{β (p - q + 1)}{2} - 1} e^{- λ_{j} / 2},

as these λ_j values are the squared singular values of p × q i.i.d. Gaussian matrices. Equivalently, the eigenvalues of the matrix $A^{†} A \in F^{q \times q}$ are frequently used for sampling purpose, where † is the conjugate transposition. The tangent spaces of noncompact symmetric spaces of the types BDI, AIII, and CII are

\{[\begin{matrix} 0 & X \\ X^{†} & 0 \end{matrix}] : X is p \times q matrix\},

(9.1)

and a choice of maximal Abelian subalgebra $a$ is the set with X being (nonsquare) diagonal matrix with diagonal elements h₁, …, h_q. The KAK decomposition G = KAK of the noncompact symmetric spaces BDI, AIII, and CII is the hyperbolic CS decomposition (HCSD).^71,72 The decomposition $p = \cup_{k \in K} k a k^{- 1}$ is the p × q SVD on upper right p × q corner. The restricted roots are the following (β = 1, 2, 4):

(9.2)

A. Noncompact BDI, β = 1 Laguerre

The noncompact symmetric space type BDI is G/K = O(p, q)/(O(p) × O(q)). The tangent space $p$ (9.1) has the Gaussian measure as i.i.d. Gaussian distribution endowed on the elements of X. For $M \in p$ ⁠, it is $\exp (- t r (M^{T} M)) d p$ ⁠. From (2.7) using (9.2) β = 1, we obtain

\exp (- t r (M^{T} M)) d p \propto \prod_{j < k} | λ_{j} - λ_{k} | \prod_{j = 1}^{q} e^{- λ_{j} / 2} λ_{j}^{\frac{p - q - 1}{2}} d λ_{1} \dots d λ_{q},

with the change of variables $λ_{j} = h_{j}^{2}$ ⁠. Thus, the values λ₁, …, λ_q are the squared singular values of the upper right corner of M. The obtained measure is the joint density of the β = 1 Laguerre ensemble.

B. Noncompact AIII, β = 2 Laguerre

The noncompact symmetric space type AIII is G/K = U(p, q)/(U(p) × U(q)). The tangent space (9.1) has the Gaussian measure as i.i.d. complex Gaussian distribution endowed on the elements of X. For $M \in p$ ⁠, that is $\exp (- t r (M^{H} M)) d p$ ⁠. From (2.7) and using (9.2) β = 2, we obtain

\exp (- t r (M^{H} M)) d p \propto \prod_{j < k} | λ_{j} - λ_{k} |^{2} \prod_{j = 1}^{q} e^{- λ_{j} / 2} λ_{j}^{p - q} d λ_{1} \dots d λ_{q},

with the change of variables $λ_{j} = h_{j}^{2}$ ⁠. Again, the values λ₁, …, λ_q are the squared singular values of the upper right corner of M. The obtained measure is the joint density of the β = 2 Laguerre ensemble.

C. Noncompact CII, β = 4 Laguerre

The noncompact symmetric space CII is $G / K = U (p, q, H) / (U (p, H) \times U (q, H))$ ⁠. The tangent space (9.1) has the Gaussian measure as i.i.d. quaternionic Gaussian distribution endowed on the elements of X. For $M \in p$ ⁠, that is $\exp (- t r (M^{D} M)) d p$ ⁠. From (2.7) and using (9.2) β = 4, we obtain

\exp (- t r (M^{D} M)) d p \propto \prod_{j < k} | λ_{j} - λ_{k} |^{4} \prod_{j = 1}^{q} e^{- λ_{j} / 2} λ_{j}^{2 (p - q) + 1} d λ_{1} \dots d λ_{q},

with the change of variables $λ_{j} = h_{j}^{2}$ ⁠. The values λ₁, …, λ_q are the squared singular values of the upper right corner of M. The obtained measure is the joint density of the β = 4 Laguerre ensemble.

ACKNOWLEDGMENTS

We thank Martin Zirnbauer for the lengthy email thread from 2001, where he patiently explained which random matrix ensembles seemed to be covered by symmetric spaces. We thank Eduardo Dueñez for another lengthy email thread back in 2013. We thank Pavel Etingof for suggesting the K₁AK₂ decomposition and pointing us to key references, Bernie Wang for so very much, and the Fall 2020 Random Matrix Theory class (MIT 18.338) for valuable suggestions. We also thank Sigurður Helgason for lively discussions by email. We acknowledge NSF Grant Nos. OAC-1835443, OAC-2103804, SII-2029670, ECCS-2029670, and PHY-2021825 for financial support.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

REFERENCES

1.

F. J.

Dyson

, “

Statistical theory of the energy levels of complex systems. I

,”

J. Math. Phys.

3

(

1

),

140

–

156

(

1962

).

https://doi.org/10.1063/1.1703773

Google Scholar

2.

F. J.

Dyson

, “

Statistical theory of the energy levels of complex systems. II

,”

J. Math. Phys.

3

(

1

),

157

–

165

(

1962

).

https://doi.org/10.1063/1.1703774

Google Scholar

3.

F. J.

Dyson

, “

Statistical theory of the energy levels of complex systems. III

,”

J. Math. Phys.

3

(

1

),

166

–

175

(

1962

).

https://doi.org/10.1063/1.1703775

Google Scholar

4.

F. J.

Dyson

, “

The threefold way. Algebraic structure of symmetry groups and ensembles in quantum mechanics

,”

J. Math. Phys.

3

(

6

),

1199

–

1215

(

1962

).

https://doi.org/10.1063/1.1703863

Google Scholar

5.

F. J.

Dyson

, “

Correlations between eigenvalues of a random matrix

,”

Commun. Math. Phys.

19

(

3

),

235

–

250

(

1970

).

https://doi.org/10.1007/bf01646824

Google Scholar

6.

A.

Altland

and

M. R.

Zirnbauer

, “

Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures

,”

Phys. Rev. B

55

(

2

),

1142

(

1997

).

https://doi.org/10.1103/physrevb.55.1142

Google Scholar

7.

M. R.

Zirnbauer

, “

Riemannian symmetric superspaces and their origin in random-matrix theory

,”

J. Math. Phys.

37

(

10

),

4986

–

5018

(

1996

).

https://doi.org/10.1063/1.531675

Google Scholar

8.

M.

Caselle

, “

A new classification scheme for random matrix theories

,” arXiv:cond-mat/9610017 (

1996

).

Google Scholar

9.

D. A.

Ivanov

, “

Random-matrix ensembles in p-wave vortices

,” in

Vortices in Unconventional Superconductors and Superfluids

(

Springer

,

2002

), pp.

253

–

265

.

Google Scholar

10.

N.

Katz

and

P.

Sarnak

,

Random Matrices, Frobenius Eigenvalues, and Monodromy

(

American Mathematical Society

,

1999

), Vol. 45.

Google Scholar

11.

N.

Katz

and

P.

Sarnak

, “

Zeroes of zeta functions and symmetry

,”

Bull. Am. Math. Soc.

36

(

1

),

1

–

26

(

1999

).

https://doi.org/10.1090/s0273-0979-99-00766-1

Google Scholar

12.

E.

Dueñez

, “

Random matrix ensembles associated to compact symmetric spaces

,” Ph.D. thesis,

Princeton University

,

2001

.

Google Scholar

13.

E.

Dueñez

, “

Random matrix ensembles associated to compact symmetric spaces

,”

Commun. Math. Phys.

244

(

1

),

29

–

61

(

2004

).

https://doi.org/10.1007/s00220-003-0994-2

Google Scholar

14.

P. J.

Forrester

, “

Random matrices, log-gases and the Calogero-Sutherland model

,” in

Quantum Many-Body Problems and Representation Theory

(

Mathematical Society of Japan

,

1998

), pp.

97

–

181

.

Google Scholar

15.

P. J.

Forrester

,

Log-Gases and Random Matrices

(

Princeton University Press

,

2010

).

Google Scholar

16.

G. W.

Anderson

,

A.

Guionnet

, and

O.

Zeitouni

,

An Introduction to Random Matrices

(

Cambridge University Press

,

2010

), Vol. 118.

Google Scholar

17.

A.

Edelman

and

N. R.

Rao

, “

Random matrix theory

,”

Acta Numer.

14

,

233

–

297

(

2005

).

https://doi.org/10.1017/s0962492904000236

Google Scholar

18.

C. C.

Paige

and

M. A.

Saunders

, “

Towards a generalized singular value decomposition

,”

SIAM J. Numer. Anal.

18

(

3

),

398

–

405

(

1981

).

https://doi.org/10.1137/0718026

Google Scholar

19.

C. F.

Van Loan

, “

Generalizing the singular value decomposition

,”

SIAM J. Numer. Anal.

13

(

1

),

76

–

83

(

1976

).

https://doi.org/10.1137/0713009

Google Scholar

20.

A.

Edelman

and

B. D.

Sutton

, “

The beta-Jacobi matrix model, the CS decomposition, and generalized singular value problems

,”

Found. Comput. Math.

8

(

2

),

259

–

285

(

2008

).

https://doi.org/10.1007/s10208-006-0215-9

Google Scholar

21.

A.

Edelman

and

Y.

Wang

, “

Random hyperplanes, generalized singular values & ‘what’s my β?

’,” in

2018 IEEE Statistical Signal Processing Workshop (SSP)

(

IEEE

,

2018

), pp.

458

–

462

.

Google Scholar

22.

M.

Flensted-Jensen

, “

Spherical functions on a real semisimple Lie group. A method of reduction to the complex case

,”

J. Funct. Anal.

30

(

1

),

106

–

146

(

1978

).

https://doi.org/10.1016/0022-1236(78)90058-7

Google Scholar

23.

B.

Hoogenboom

,

The Generalized Cartan Decomposition for a Compact Lie Group

(

Stichting Mathematisch Centrum. Zuivere Wiskunde

,

1983

).

Google Scholar

24.

A.

Edelman

and

Y.

Wang

, “

The GSVD: Where are the ellipses?, matrix trigonometry, and more

,”

SIAM J. Matrix Anal. Appl.

41

(

4

),

1826

–

1856

(

2020

).

https://doi.org/10.1137/18m1234412

Google Scholar

25.

Alternatively, one can imagine the partial format of the CS decomposition. This is also equivalent to the bi-Stiefel decomposition with another quotient by the orthogonal group on the right.

26.

I.

Dumitriu

and

A.

Edelman

, “

Matrix models for beta ensembles

,”

J. Math. Phys.

43

(

11

),

5830

–

5847

(

2002

).

https://doi.org/10.1063/1.1507823

Google Scholar

27.

R.

Killip

and

I.

Nenciu

, “

Matrix models for circular ensembles

,”

Int. Math. Res. Not.

2004

(

50

),

2665

–

2701

.

https://doi.org/10.1155/s1073792804141597

28.

S.

Helgason

,

Groups & Geometric Analysis: Radon Transforms, Invariant Differential Operators and Spherical Functions: Volume 1

(

Academic Press

,

1984

).

Google Scholar

29.

E.

Cartan

, “

Sur une classe remarquable d’espaces de Riemann

,”

Bull. Soc. Math. Fr.

54

,

214

–

264

(

1926

).

https://doi.org/10.24033/bsmf.1105

Google Scholar

30.

E.

Cartan

, “

Sur certaines formes Riemanniennes remarquables des géométries à groupe fondamental simple

,”

Ann. Sci. Ec. Norm. Super.

44

,

345

–

467

(

1927

).

https://doi.org/10.24033/asens.781

Google Scholar

31.

E.

Cartan

, “

Sur une classe remarquable d’espaces de Riemann. II

,”

Bull. Soc. Math. Fr.

55

,

114

–

134

(

1927

).

https://doi.org/10.24033/bsmf.1113

Google Scholar

32.

E.

Cartan

, “

Sur la détermination d’un système orthogonal complet dans un espace de Riemann symétrique clos

,”

Rend. Circolo Mat. Palermo

53

(

1

),

217

–

252

(

1929

).

https://doi.org/10.1007/bf03024106

Google Scholar

33.

S.

Helgason

,

Differential Geometry, Lie Groups, and Symmetric Spaces

(

Academic Press

,

1978

).

Google Scholar

34.

J.

Segel

,

Recountings: Conversations with MIT Mathematicians

(

CRC Press

,

2009

), http://www-math.mit.edu/helgason/helgason_interview.pdf.

Google Scholar

35.

A.

Hurwitz

, “

Ueber die erzeugung der invarianten durch integration

,” in

Mathematische Werke

(

Springer

,

1963

), pp.

546

–

564

.

Google Scholar

36.

H.

Weyl

,

The Classical Croups: Their Invariants and Representations

(

Princeton University Press

,

1946

), Vol. 45.

Google Scholar

37.

In Ref. 73, Mehta credits Hsu⁴² for the GOE. In fact, Ref. 42 has the Jacobian for the symmetric real eigenvalue problem and indeed works with AA^T where A = randn(m,n) but does not work with A + A^T. It is no doubt Hsu⁴² could have instantly written down the GOE distribution if he had only been asked.

38.

E. P.

Wigner

, “

Characteristic vectors of bordered matrices with infinite dimensions

,”

Ann. Math.

62

,

548

–

564

(

1955

).

https://doi.org/10.2307/1970079

Google Scholar

39.

E. P.

Wigner

, “

On the distribution of the roots of certain symmetric matrices

,”

Ann. Math.

67

,

325

–

327

(

1958

).

https://doi.org/10.2307/1970008

Google Scholar

40.

R. A.

Fisher

, “

The sampling distribution of some statistics obtained from non-linear equations

,”

Ann. Eugen.

9

(

3

),

238

–

249

(

1939

).

https://doi.org/10.1111/j.1469-1809.1939.tb02211.x

Google Scholar

41.

S. N.

Roy

, “

p-Statistics or some generalisations in analysis of variance appropriate to multivariate problems

,”

Sankhyā

4

,

381

–

396

(

1939

).

Google Scholar

42.

P. L.

Hsu

, “

On the distribution of roots of certain determinantal equations

,”

Ann. Eugen.

9

(

3

),

250

–

258

(

1939

).

https://doi.org/10.1111/j.1469-1809.1939.tb02212.x

Google Scholar

43.

H.

Leff

, “

Statistical theory of energy-level spacing distributions for complex spectra

,” Ph.D. thesis,

University of Iowa

,

1963

.

Google Scholar

44.

M.

Flensted-Jensen

, “

Discrete series for semisimple symmetric spaces

,”

Ann. Math.

111

,

253

–

311

(

1980

).

https://doi.org/10.2307/1971201

Google Scholar

45.

T.

Matsuki

, “

Double coset decompositions of algebraic groups arising from two involutions I

,”

J. Algebra

175

(

3

),

865

–

925

(

1995

).

https://doi.org/10.1006/jabr.1995.1218

Google Scholar

46.

T.

Matsuki

, “

Double coset decompositions of reductive Lie groups arising from two involutions

,”

J. Algebra

197

(

1

),

49

–

91

(

1997

).

https://doi.org/10.1006/jabr.1997.7123

Google Scholar

47.

T.

Matsuki

, “

Classification of two involutions on compact semisimple Lie groups and root systems

,”

J. Lie Theory

12

(

1

),

41

–

68

(

2002

).

Google Scholar

48.

T.

Kobayashi

, “

A generalized Cartan decomposition for the double coset space (U(n₁) × U(n₂) × U(n₃))\U(n)/(U(p) × U(q))

,”

J. Math. Soc. Jpn.

59

(

3

),

669

–

691

(

2007

).

https://doi.org/10.2969/jmsj/05930669

Google Scholar

49.

B.

Hoogenboom

,

Intertwining Functions on Compact Lie Groups, I

(

Stichting Mathematisch Centrum. Zuivere Wiskunde

,

1983

).

Google Scholar

50.

A. T.

James

and

A. G.

Constantine

, “

Generalized Jacobi polynomials as spherical functions of the Grassmann manifold

,”

Proc. London Math. Soc.

s3-29

(

1

),

174

–

192

(

1974

).

https://doi.org/10.1112/plms/s3-29.1.174

Google Scholar

51.

D.

Bump

,

Lie Groups

(

Springer

,

2004

).

Google Scholar

52.

R.

Gilmore

,

Lie Groups, Lie Algebras, and Some of Their Applications

(

Courier Corporation

,

2012

).

Google Scholar

53.

A. W.

Knapp

,

Lie Groups Beyond an Introduction

(

Springer Science & Business Media

,

2013

), Vol. 140.

Google Scholar

54.

A. A.

Kirillov

,

Representation Theory and Noncommutative Harmonic Analysis II: Homogeneous Spaces, Representations and Special Functions

(

Springer

,

1995

).

Google Scholar

55.

A. W.

Knapp

,

Representation Theory of Semisimple Groups: An Overview Based on Examples

(

Princeton University Press

,

2001

), Vol. 36.

Google Scholar

56.

R.

Hermann

, “

Variational completeness for compact symmetric spaces

,”

Proc. Am. Math. Soc.

11

(

4

),

544

–

546

(

1960

).

https://doi.org/10.1090/s0002-9939-1960-0124436-6

Google Scholar

57.

A.

Kollross

, “

A classification of hyperpolar and cohomogeneity one actions

,”

Trans. Am. Math. Soc.

354

(

2

),

571

–

612

(

2002

).

https://doi.org/10.2307/2693761

Google Scholar

58.

M. R.

Zirnbauer

and

F. D. M.

Haldane

, “

Single-particle Green’s functions of the Calogero-Sutherland model at couplings λ = 1/2, 1, and 2

,”

Phys. Rev. B

52

(

12

),

8729

(

1995

).

https://doi.org/10.1103/physrevb.52.8729

Google Scholar

59.

M.

Caselle

and

U.

Magnea

, “

Random matrix theory and symmetric spaces

,”

Phys. Rep.

394

(

2–3

),

41

–

156

(

2004

).

https://doi.org/10.1016/j.physrep.2003.12.004

Google Scholar

60.

The actual development of the Jacobians (2.2), (2.3) was done the other way around. In Ref. 61, Helgason credits Cartan³² for the derivation of these Jacobians, which was then computed only for symmetric spaces. The KAK decomposition was discovered later in 1950s, and the Jacobians are identically extended from the decomposition of G/K to the decomposition of G.

61.

S.

Helgason

,

Differential Geometry and Symmetric Spaces

(

Academic Press

,

1962

).

Google Scholar

62.

A.

Terras

,

Harmonic Analysis on Symmetric Spaces—Higher Rank Spaces, Positive Definite Matrix Space and Generalizations

(

Springer

,

2016

).

Google Scholar

63.

J.

An

,

Z.

Wang

, and

K.

Yan

, “

A generalization of random matrix ensemble, I: General theory

,”

Pacific J. Math.

228

(

1

),

1

–

17

(

2006

).

https://doi.org/10.2140/pjm.2006.228.1

Google Scholar

64.

M. L.

Mehta

,

Random Matrices

(

Elsevier

,

2004

).

Google Scholar

65.

S.

Heuveline

,

S.

Said

, and

C.

Mostajeran

, “

Gaussian distributions on Riemannian symmetric spaces, random matrices, and planar Feynman diagrams

,” arXiv:2106.08953 (

2021

).

Google Scholar

66.

R.

Onn

,

A. O.

Steinhardt

, and

A.

Bojanczyk

, “

The hyperbolic singular value decomposition and applications

,” in

Proceedings of the 32nd Midwest Symposium on Circuits and Systems

(

IEEE

,

1989

), pp.

575

–

577

.

Google Scholar

67.

H.

Führ

and

Z.

Rzeszotnik

, “

A note on factoring unitary matrices

,”

Linear Algebra Appl.

547

,

32

–

44

(

2018

).

https://doi.org/10.1016/j.laa.2018.02.017

Google Scholar

68.

G.

Blower

,

Random Matrices: High Dimensional Phenomena

(

Cambridge University Press

,

2009

), Vol. 367.

Google Scholar

69.

C.

Davis

and

W. M.

Kahan

, “

Some new bounds on perturbation of subspaces

,”

Bull. Am. Math. Soc.

75

(

4

),

863

–

868

(

1969

).

https://doi.org/10.1090/s0002-9904-1969-12330-x

Google Scholar

70.

C.

Davis

and

W. M.

Kahan

, “

The rotation of eigenvectors by a perturbation. III

,”

SIAM J. Numer. Anal.

7

(

1

),

1

–

46

(

1970

).

https://doi.org/10.1137/0707001

Google Scholar

71.

E. J.

Grimme

,

D. C.

Sorensen

, and

P.

Van Dooren

, “

Model reduction of state space systems via an implicitly restarted Lanczos method

,”

Numer. Algorithms

12

(

1

),

1

–

31

(

1996

).

https://doi.org/10.1007/bf02141739

Google Scholar

72.

N. J.

Higham

, “

J-orthogonal matrices: Properties and generation

,”

SIAM Rev.

45

(

3

),

504

–

519

(

2003

).

https://doi.org/10.1137/s0036144502414930

Google Scholar

73.

M. L.

Mehta

, “

On the statistical properties of the level-spacings in nuclear spectra

,”

Nuclear Physics

18

,

395

–

419

(

1960

).

Google Scholar

2022

Author(s)

On the Cartan decomposition for classical random matrix ensembles

I. INTRODUCTION

A. Classical random matrix ensembles

B. Coordinate systems on the Grassmannian manifold

C. Contributions of this paper

II. BACKGROUND

A. Joint densities of classical random matrix ensembles

B. Symmetric space and the generalized Cartan decomposition

C. A symmetric space: one RMT or many RMTs?

III. CARTAN’S IDEA: A MODERNIZED APPROACH

A. The ping pong operator, ping pong vectors, and ping pong subspaces

B. The Kronecker product, linear operator adX, and its exponential

C. Antisymmetric and symmetric matrices: An important first example of symmetric space as ping pong spaces

D. General k and p arise from an involution θ

E. The operators adH,eadH⁠, and the subspaces k,p

F. Symmetric spaces

G. When G is a compact Lie group

H. Jacobian of the map Φ

I. Extension to the generalized Cartan decomposition

IV. RANDOM MATRIX ENSEMBLES: COMPACT AND NONCOMPACT

A. Compact symmetric spaces

B. Noncompact symmetric spaces

C. Non-probability measure of noncompact groups

V. COMPACT AI, A, and AII: CIRCULAR ENSEMBLES

A. Compact AI, β=1 COE

B. Compact A, β=2 CUE

C. Compact AII, β=4 CSE

VI. COMPACT BDI, AIII, and CII: JACOBI ENSEMBLES

A. Compact BDI-I, β=1 Jacobi

B. Compact AIII-III, β = 2 Jacobi

C. Compact CII-II, β = 4 Jacobi

VII. COMPACT MIXED TYPES: MORE CIRCULAR AND JACOBI

A. Compact AI-II

B. Compact AI-III, CI-II

C. Compact DI-III, AII-III

VIII. NONCOMPACT AI, A, AND AII: HERMITE ENSEMBLES

A. Noncompact AI, β = 1 GOE

B. Noncompact A, β = 2 GUE

C. Noncompact AII, β = 4 GSE

IX. NONCOMPACT BDI, AIII, and CII: LAGUERRE ENSEMBLES

A. Noncompact BDI, β = 1 Laguerre

B. Noncompact AIII, β = 2 Laguerre

C. Noncompact CII, β = 4 Laguerre

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

DATA AVAILABILITY

REFERENCES

Citing articles via

Submit your article

Sign up for alerts

Related Content

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

This Feature Is Available To Subscribers Only

B. The Kronecker product, linear operator ad_X, and its exponential

D. General $k$ and $p$ arise from an involution $θ$

E. The operators ${a d}_{H}, e^{{a d}_{H}}$ ⁠, and the subspaces $k, p$

A. Compact AI, $β = 1$ COE

B. Compact A, $β = 2$ CUE

C. Compact AII, $β = 4$ CSE

A. Compact BDI-I, $β = 1$ Jacobi