Congruence and Conjunctivity of Matrices

Abstract

Congruence of arbitrary square matrices over an arbitrary field is treated here by elementary classical methods, and likewise for conjunctivity of arbitrary square matrices over an arbitrary field with involution. Uniqueness results are emphasized, since they are largely neglected in the literature. In particular, it is shown that a matrix S is congruent [conjunctive] to S 0⊕ S 1 with S 1 nonsingular, and that if S 1 here is of maximal size among all nonsingular matrices R 1 for which R 0⊕ R 1 is congruent [conjunctive] to S, then the congruence [conjunctivity] class of S determines that of S 1. Partially canonical forms (most of them already known) are derived, to the extent that they do not depend on the field. Nearly canonical forms are derived for “neutral” matrices (those congruent or conjunctive with block matrices O N M O with the two zero blocks being square). For a neutral matrix S over a field F,the F-congruence [ F-conjunctivity] class of S is determined by the F-equivalence class of the pencil S+ tS' [ S+ tS ∗] and, if the pencil is nonsingular, by the F[ t]-equivalence class of S+ tS' [ S+ tS ∗].