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

Đocović and Szechtman [D.Ž. Đocović, F. Szechtman, Characterization of bilinear spaces with unimodular isometry group, 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 et al. [E.S. Coakley, F.M. Dopico, C.R. Johnson, Matrices with orthogonal groups admitting only determinant one, Linear Algebra Appl. 428 (2008) 796–813] gave another proof of this criterion over R and C 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 (MT,M) for equivalence, and of M-TM (if M is nonsingular) for similarity.