Linear Algebra and its Applications 302–303 (1999) 411–421 www.elsevier.com/locate/laa The Jordan Canonical Forms of complex orthogonal and skew-symmetric matrices Roger A. Horn a,∗ , Dennis I. Merino b a Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT 84112–0090, USA b Department of Mathematics, Southeastern Louisiana University, Hammond, LA 70402-0687, USA Received 25 August 1999; accepted 3 September 1999 Submitted by B. Cain Dedicated to Hans Schneider Abstract We study the Jordan Canonical Forms of complex orthogonal and skew-symmetric matrices, and consider some related results. © 1999 Elsevier Science Inc. All rights reserved. Keywords: Canonical form; Complex orthogonal matrix; Complex skew-symmetric matrix 1. Introduction and notation Every square complex matrix A is similar to its transpose, A T ([2, Section 3.2.3] or [1, Chapter XI, Theorem 5]), and the similarity class of the n-by-n complex symmetric matrices is all of M n [2, Theorem 4.4.9], the set of n-by-n complex matrices. However, other natural similarity classes of matrices are non-trivial and can be characterized by simple conditions involving the Jordan Canonical Form. For example, A is similar to its complex conjugate, A (and hence also to its adjoint, A ∗ = A T ), if and only if A is similar to a real matrix [2, Theorem 4.1.7]; the Jordan Canonical Form of such a matrix can contain only Jordan blocks with real eigenvalues and pairs of Jordan blocks of the form J k (λ) ⊕ J k ( λ) for non-real λ. We denote by J k (λ) the standard upper triangular k-by-k Jordan block with eigenvalue ∗ Corresponding author. E-mail addresses: rhorn@math.utah.edu (R.A. Horn), fmat1649@selu.edu (D.I. Merino) 0024-3795/99/$ - see front matter  1999 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 4 - 3 7 9 5 ( 9 9 ) 0 0 1 9 9 - 8