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A unifying approach is presented between similarity, consimilarity, T congruence and *congruence of a matrix A to a symmetric, to a Hermitian or to a real matrix. Also studied are similarity consimilarity, T congruence and *congruence of a matrix A to A∗, AT , and A. 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 AA is nonderogatory and nonsingular, then the consimilarities between A and AT are precisely the Hermitian similarities between AA and (AA)∗. Also, if A is nonsingular, then the coninvolutory T congruences between A and A∗ are in 1-to-1 correspondence with Hermitian similarities between A(A−1)T and (A(A−1)T )∗.

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