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

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

2010-04-21

arxiv.org/abs/1004.3603v1

Linear Algebra Appl. 431 (2009) 1620-1632

Mathematics

Representation Theory

21 pages

Scientific paper

10.1016/j.laa.2009.05.034

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.

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