2009/07/12 by Tatyana G. Gerasimova, Roger A. Horn, Vladimir V. Sergeichuk
Computer Science · Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Matrix Theory and Algorithms #math.RT #msc:15A21 #msc:15A63
paper · pdf · doi:10.1016/j.laa.2009.05.034
published as Linear Algebra Appl. 431 (2009) 1620-1632 · 21 pages
openalex publication_date 2009/07/12 · arxiv created 2010/04/21 · arxiv updated 2010/04/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 (MT,M) for equivalence, and of M-TM (if M is nonsingular) for similarity.