Solving the Problem of Simultaneous Diagonalization of Complex Symmetric Matrices via Congruence
Miguel D. Bustamante, Pauline Mellon, M. Victoria Velasco · SIAM Journal on Matrix Analysis and Applications · 2020
We provide a solution to the problem of simultaneous diagonalization via congruence of a given set of $m$ complex symmetric $n\times n$ matrices $\{A_{1},\ldots,A_{m}\}$, 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 diagonalization via similarity 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.