On (con)similarities and congruences between A and A^*, A^T or A

Abstract

A unifying approach is presented between similarity, consimilarity, ^Tcongruence and *congruence of a matrix A to a symmetric, to a Hermitian or to a real matrix. Also studied are similarity consimilarity, ^Tcongruence and *congruence of a matrix A to A^∗, A^T , and Ā. Attempts are made to find special (con)similarities and congruences, as well as to find connections between these classes of maps. For example, it is shown that if AĀ is nonderogatory and nonsingular, then the consimilarities between A and A^T are precisely the Hermitian similarities between AĀ and (AĀ)^∗. Also, if A is nonsingular, then the coninvolutory ^Tcongruences between A and A^∗ are in 1-to-1 correspondence with Hermitian similarities between A(A⁻¹)^T and (A(A⁻¹)^{T )∗}.