Geodesic
On the Solutions of the Matrix Equation f(X,X *)=g(X,X *)
Canadian mathematical bulletin, Tome 15 (1972) no. 1, pp. 45-49

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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. 
Basavappa, P. On the Solutions of the Matrix Equation f(X,X *)=g(X,X *). Canadian mathematical bulletin, Tome 15 (1972) no. 1, pp. 45-49. doi: 10.4153/CMB-1972-010-9
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     title = {On the {Solutions} of the {Matrix} {Equation} {f(X,X} {*)=g(X,X} *)},
     journal = {Canadian mathematical bulletin},
     pages = {45--49},
     year = {1972},
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     number = {1},
     doi = {10.4153/CMB-1972-010-9},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1972-010-9/}
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