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

## TL;DR

This paper presents a finite-step procedure to determine whether a set of complex symmetric matrices can be simultaneously diagonalized via congruence, solving a longstanding problem in the complex case.

## Contribution

It reduces the simultaneous diagonalization via congruence problem to a related similarity problem, providing a practical finite-step solution.

## Key findings

- Provides a finite-step procedure for diagonalization via congruence.
- Reduces the problem to a classical similarity diagonalization problem.
- Solves a long-standing open problem in the complex case.

## Abstract

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.