A Normal form for a Matrix under the Unitary Congruence Group
Dante C. Youla · Canadian Journal of Mathematics · 1961
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 μ 1 2 , μ 2 2 , … , μ n 2 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 α 1 2 , α 2 2 , … , α k 2 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.