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arXiv · 1508.05859

Matrix exponentials, SU(N) group elements, and real polynomial roots

T. S. Van Kortryk
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Abstract

The exponential of an NxN 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 NxN 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 (N-1)-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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T. S. Van Kortryk. 2015-09-01. Matrix exponentials, SU(N) group elements, and real polynomial roots. https://doi.org/10.1063/1.4938418

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