Solving the problem of simultaneous diagonalization of complex symmetric matrices via congruence

Miguel D. Bustamante; Pauline Mellon; M. Victoria Velasco

doi:10.1137/19M1280430

paper

Solving the problem of simultaneous diagonalization of complex symmetric matrices via congruence

arXiv:1908.04228 · doi:10.1137/19M1280430

Abstract

We provide a solution to the problem of simultaneous $d ia g o na l i z a t i o n$ $v ia$ $co n g r u e n ce$ of a given set of $m$ complex symmetric $n \times n$ matrices ${A_{1}, \dots, A_{m}}$ , by showing that it can be reduced to a possibly lower-dimensional problem where the question is rephrased in terms of the classical problem of simultaneous $d ia g o na l i z a t i o n$ $v ia$ $s imi l a r i t y$ of a new related set of matrices. We provide a procedure to determine in a finite number of steps whether or not a set of matrices is simultaneously diagonalizable by congruence. This solves a long standing problem in the complex case.