The exponential of an N × N matrix can always be expressed as a matrix polynomial of order N − 1. In particular, a general group element for the fundamental representation of SU(N) can be expressed as a matrix polynomial of order N − 1 in a traceless N × N hermitian generating matrix, with polynomial coefficients consisting of elementary trigonometric functions dependent on N − 2 invariants in addition to the group parameter. These invariants are just angles determined by the direction of a real N-vector whose components are the eigenvalues of the hermitian matrix. Equivalently, the eigenvalues are given by projecting the vertices of an -simplex onto a particular axis passing through the center of the simplex. The orientation of the simplex relative to this axis determines the angular invariants and hence the real eigenvalues of the matrix.
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The same procedure may be used to construct sequentially a hierarchy of polynomials for the exponentials of almost any matrix, as in (1).
Although the eigenvalues themselves still define a circle with the same radius as before, namely r2, since .
Here is the solution to the exercise: The tetrahedron’s vertices are at a distance from the origin, such that , , , and , where .