A Normal form for a Matrix under the Unitary Congruence Group

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, … , μ_n² 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, … , α_k² 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.