Congruences of a square matrix and its transpose

被引：53

作者：

Horn, RA

Sergeichuk, VV

机构：

[1] Univ Utah, Dept Math, Salt Lake City, UT 84103 USA

[2] Ukrainian Acad Sci, Inst Math, Kiev, Ukraine

来源：

LINEAR ALGEBRA AND ITS APPLICATIONS | 2004年 / 389卷

基金：

美国国家科学基金会;

关键词：

congruence; sesquilinear forms; canonical forms;

D O I：

10.1016/j.laa.2004.03.010

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

It is known that any square matrix A over any field is congruent to its transpose: A(T) = S-T AS for some nonsingular S; moreover, S can be chosen such that S-2 = I, that is, S can be chosen to be involutory. We show that A and A(T) are *congruent over any field F of characteristic not two with involution a --> (a) over bar (the involution can be the identity): A(T) = (S) over bar (T) AS for some nonsingular S; moreover, S can be chosen such that (S) over barS = 1, that is, S can be chosen to be coninvolutory. The short and simple proof is based on Sergeichuk's canonical form for *congruence [Math. USSR, Izvestiya 31 (3) (1988) 481]. It follows that any matrix A over F can be represented as A = EB, in which E is coninvolutory and B is symmetric. (C) 2004 Elsevier Inc. All rights reserved.

引用

页码：347 / 353

页数：7

共 8 条

[1] A square matrix is congruent to its transpose [J].