Matrices that are self-congruent only via matrices of determinant one

TL;DR

This paper provides a new proof and characterization of matrices that are only self-congruent via matrices of determinant one, extending previous results on isometries and canonical forms over various fields.

Contribution

It introduces a novel proof using canonical matrices for congruence and establishes necessary and sufficient conditions involving canonical forms for congruence, equivalence, and similarity.

Findings

New proof of the criterion over fields using canonical matrices

Conditions involving canonical forms for congruence, equivalence, and similarity

Extension of previous results to broader matrix classes

Abstract

Docovic and Szechtman, [Proc. Amer. Math. Soc. 133 (2005) 2853-2863] considered a vector space V endowed with a bilinear form. They proved that all isometries of V over a field F of characteristic not 2 have determinant 1 if and only if V has no orthogonal summands of odd dimension (the case of characteristic 2 was also considered). Their proof is based on Riehm's classification of bilinear forms. Coakley, Dopico, and Johnson [Linear Algebra Appl. 428 (2008) 796-813] gave another proof of this criterion over the fields of real and complex numbers using Thompson's canonical pairs of symmetric and skew-symmetric matrices for congruence. Let M be the matrix of the bilinear form on V. We give another proof of this criterion over F using our canonical matrices for congruence and obtain necessary and sufficient conditions involving canonical forms of M for congruence, of (M^T,M) for equivalence, and of M^{-T}M (if M is nonsingular) for similarity.