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

Voir la notice de l'article provenant de la source Cambridge University Press

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
@article{10_4153_CMB_1972_010_9,
     author = {Basavappa, P.},
     title = {On the {Solutions} of the {Matrix} {Equation} {f(X,X} {*)=g(X,X} *)},
     journal = {Canadian mathematical bulletin},
     pages = {45--49},
     year = {1972},
     volume = {15},
     number = {1},
     doi = {10.4153/CMB-1972-010-9},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1972-010-9/}
}
TY  - JOUR
AU  - Basavappa, P.
TI  - On the Solutions of the Matrix Equation f(X,X *)=g(X,X *)
JO  - Canadian mathematical bulletin
PY  - 1972
SP  - 45
EP  - 49
VL  - 15
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1972-010-9/
DO  - 10.4153/CMB-1972-010-9
ID  - 10_4153_CMB_1972_010_9
ER  - 
%0 Journal Article
%A Basavappa, P.
%T On the Solutions of the Matrix Equation f(X,X *)=g(X,X *)
%J Canadian mathematical bulletin
%D 1972
%P 45-49
%V 15
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-1972-010-9/
%R 10.4153/CMB-1972-010-9
%F 10_4153_CMB_1972_010_9

[1] 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] 2. Gantmacher, F. R., The theory of matrices, Vol. I (translated from Russian). Chelsea, New York, 1959. Google Scholar

[3] 3. Weitzenböck, R., Über die Matrixgleichung XX' = A., Proc. Akad. Wet. Amsterdam. 35 (1932), 328-330. Google Scholar

Cité par Sources :