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A Normal form for a Matrix under the Unitary Congruence Group

Published online by Cambridge University Press:  20 November 2018

D. C. Youla
D. C. Youla*
Affiliation:
Polytechnic Institute of Brooklyn
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Let C be a square matrix with complex elements. If C = C' (C' denotes the transpose of C) there exists a unitary matrix U such that

(1)

where the μ's are the non-negative square roots of the eigenvalues μ12, μ22, … , μn2 of C*C (C* is the adjoint of C) (2). If C is skew-symmetric, that is, C= — C”, there exists a unitary matrix U such that

(2)

where

(3)

and the α's are the positive square roots of the non-zero eigenvalues α12, α22, … , αk2 of C*C (1). Clearly rank C = 2k and the number of zeros appearing in (2) is n — 2k. Both (1) and (2) are classical. In a recent paper (3) Stander and Wiegman, apparently unaware of (1), give an alternative derivation of (2) with its appropriate generalization to quaternions.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 13 , 1961 , pp. 694 - 704
Copyright
Copyright © Canadian Mathematical Society 1961

References

1. Hua, L. K., On the theory of automorphic functions of a matrix variable I-geometrical basis, Amer. J. Math., 66 (1944).Google Scholar
2. Schur, I., Kin Satz ueber quadratische Formen mit komplexen Koeffizienten, Amer. J. Math., 67 (1942), 472.Google Scholar
3. Stander, J. W. and Wiegman, N. A., Canonical forms for certain matrices under unitary congruence, Can. J. Math., 12 (1960).Google Scholar
4. Youla, D. C., Direct single frequency synthesis from a prescribed scattering matrix, IRE Transactions of the Professional Group on Circuit Theory, vol. CT-6, no. 4 (December, 1959).Google Scholar
5. Youla, D. C., Some new and useful matrix results, Mem. 34, Polytechnic Institute of Brooklyn, Microwave Research Institute (March 7, 1960).Google Scholar