Almost Skew-Symmetric Matrices
Judith J. McDonald, Panayiotis Psarrakos, Michael J. Tsatsomeros · Rocky Mountain Journal of Mathematics · 2004
ABSTRACT. Almost skew-symmetric matrices are real matrices whose symmetric parts have rank one. Using the notion of the numerical range, we obtain eigenvalue inequalities and a localization of the spectrum of an almost skew-symmetric matrix. We show that almost skew-symmetry is invariant under principal pivot transformation and inversion, and that the symmetric parts of Schur complements in almost skewsymmetric matrices have rank at most one. We also use affine combinations of A and At to gain further insight into eigenvalue location and the numerical range of an almost skewsymmetric matrix. 1. Introduction. Let Mn(R) (Mn(C)) be the algebra of all n × n real (complex) matrices. In this article, we consider matrices A ∈Mn(R), n ≥ 2, whose symmetric parts have rank one. This means that the spectrum of the symmetric part of such a matrix A consists of