仅通过行列式1的矩阵自相合的矩阵
Matrices that are self-congruent only via matrices of determinant one
摘要：
Docovic和Szechtman，[Proc.Amer.Math.Soc.133（2005）2853-2863]考虑了具有双线性型的向量空间V。他们证明了特征不为2的域F上V的所有等距都有行列式1当且仅当V没有奇数维的正交和数（也考虑了特征为2的情况）。他们的证明是基于Riehm的分类双线性型。Coakley、Dopico和Johnson [线性代数Appl.428（2008）796-813]使用Thompson的对称矩阵和反对称矩阵的标准对的同余，给出了实数和复数域上的该准则的另一个证明。设M是V上双线性型的矩阵，我们利用我们的标准同余矩阵给出了F上这一准则的另一个证明，并得到了M的标准同余、（M^T，M）的标准等价和M^{-T}M（若M非奇异）的标准相似的充要条件.
英文摘要：
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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