Congruence and Conjunctivity of Matrices

AbstractCongruence 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 S0⊕S1 with S1 nonsingular, and that if S1 here is of maximal size among all nonsingular matrices R1 for which R0⊕R1 is congruent [conjunctive] to S, then the congruence [conjunctivity] class of S determines that of S1. 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 ONMO 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∗]