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

Published online by Cambridge University Press:  20 November 2018

P. Basavappa
P. Basavappa*
Affiliation:
University of North Carolina at Charlotte, Charlotte, North Carolina
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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.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 15 , Issue 1 , March 1972 , pp. 45 - 49
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Eckart, C., and Young, G., A principal axis transformation for non-hermitian matrices, Bull. Amer. Math. Soc. 45 (1939), 118-121.Google Scholar
2. Gantmacher, F. R., The theory of matrices, Vol. I (translated from Russian). Chelsea, New York, 1959.Google Scholar
3. Weitzenböck, R., Über die Matrixgleichung XX' = A., Proc. Akad. Wet. Amsterdam. 35 (1932), 328-330.Google Scholar