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...

Similarity and consimilarity of elements in the real quaternion, octonion, and sedenion algebras, as well as in the general real Cayley-Dickson algebras are considered by solving the two fundamental equations ax = xb and ax = xb in these algebras. Some consequences are also presented. AMS mathematics subject classifications: 17A05, 17A35.

in this paper, we show that a matrix a in mn(c) that has n coneigenvectors, where coneigenvaluesassociated with them are distinct, is condiagonalizable. and also show that if allconeigenvalues of conjugate-normal matrix a be real, then it is symmetric.