Unitary equivalence to a complex symmetric matrix: an algorithm

TL;DR

This paper introduces a comprehensive algorithm to determine whether a 3x3 matrix is unitarily equivalent to a complex symmetric matrix, extending the method to larger matrices with a constructive approach.

Contribution

It provides a necessary and sufficient condition for 3x3 matrices and an explicit algorithm, generalizing to almost all larger matrices, for unitary equivalence to complex symmetric matrices.

Findings

The algorithm successfully tests 3x3 matrices for unitary equivalence.

The condition is both necessary and sufficient for 3x3 matrices.

The method generalizes to almost all larger matrices.

Abstract

We present a necessary and sufficient condition for a 3 by 3 matrix to be unitarily equivalent to a symmetric matrix with complex entries, and an algorithm whereby an arbitrary 3 by 3 matrix can be tested. This test generalizes to a necessary and sufficient condition that applies to almost every n by n matrix. The test is constructive in that it explicitly exhibits the unitary equivalence to a complex symmetric matrix.