A Normal form for a Matrix under the Unitary Congruence Group
- 1 January 1961
- journal article
- Published by Canadian Mathematical Society in Canadian Journal of Mathematics
- Vol. 13, 694-704
- https://doi.org/10.4153/cjm-1961-059-8
Abstract
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.Keywords
This publication has 3 references indexed in Scilit:
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- Direct Single Frequency Synthesis from a Prescribed Scattering MatrixIRE Transactions on Circuit Theory, 1959
- Ein Satz Ueber Quadratische Formen Mit Komplexen KoeffizientenAmerican Journal of Mathematics, 1945