A note on complex matrices that are unitarily congruent to real matrices

Abstract

Every square complex matrix is known to be consimilar to a real matrix. Unitary congruence is a particular type of consimilarity. We prove that a matrix A ∈ M n ( C ) is unitarily congruent to a real matrix if and only if A is unitarily congruent to A ¯ via a symmetric unitary matrix. It is shown by an example that there exist matrices that are congruent, but not unitarily congruent, to real matrices.