对称和反对称多项式矩阵的合同标准形

Congruent Canonical Forms of Symmetric and Skew-Symmetric Polynomial Matrices

本文主要研究一元多项式环上对称矩阵和反对称矩阵的合同标准形。利用多项式和初等矩阵的性质,推广传统的理论到多项式矩阵上。通过行列式不为零的矩阵,可将一元多项式环上对称矩阵合同到三对角矩阵。由于反对称多项式矩阵的对角线元素都为零,进一步可以证明反对称多项式矩阵合同到三对角矩阵,且元素有整除性质。

This paper studies the congruent canonical forms of symmetric and skew-symmetric matrices over a univariate polynomial ring. Using the properties of polynomials and elementary matrices, the classical theory will be extended to polynomial matrices. The symmetric matrix over a univariate polynomial ring can be congruent to a tridiagonal matrix through a regular matrix. Since the diagonal elements of the skew-symmetric polynomial matrix are zero, it can be further proved that the skew-symmetric polynomial matrix is congruent to a tridiagonal matrix whose elements have the property of division.