方阵分解为两个Hermite阵乘积的充要条件

A NECESSARY AND SUFFICIENT CONDITION FOR SQUARE MATRIX TO BE DECOMPOSED AS A PRODUCT OF TWO HERMITIAN MATRICES

本文证明了下列结论: 1,设K是带有对合自同构δ的域,则域K上的方阵A可分解为两个Hermite阵乘积的充要条件是A与δ的不动域上的方阵相似。2,设K是带有对合自同构δ的域,则域K上的方阵A可分解成两个Hermite阵乘积的充要条件是A的非常数不变因子的系数全是δ的不动元。

Theorem 1. Let K be a field with an involutory automorphism δ, a neces-sary and sufficient condition for square matrix A on the field K to be deco-mposed as a product of two Hermitian matrices is that A is similar to asquare matrix on the fixed field of δ. Theorem 2. Let K be a field with an involutary automorphismδ, a nece-ssary and sufficient condition for square matrix A on the field K to be dec-omposed as a product of two Hermitian matrices is that the coefficients ofany non-constant invariant of A are fixed points of δ.