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On the Solutions of the Matrix Equation f(X,X *)=g(X,X *)

It is well known that the matrix identities XX *=I, X=X * and XX * = X * X, where X is a square matrix with complex elements, X* is the conjugate transpose of X and I is the identity matrix, characterize unitary, hermitian and normal matrices respectively. These identities are special cases of more general equations of the form (a)f(X, X *)=A and (b)f(Z, X *)=g(X, X *) where f(x, y) and g(x, y) are monomials of one of the following four forms: xyxy…xyxy, xyxy…xyx, yxyx…yxyx, and yxyx…yxy. In this paper all equations of the form (a) and (b) are solved completely. It may be noted a particular case of f(X, X *)=A, viz. XX'=A, where X is a real square matrix and X' is the transpose of X was solved by WeitzenbÖck [3]. The distinct equations given by (a) and (b) are enumerated and solved.