论文标题
多项式矩阵的半标准等效性
Semi-scalar equivalence of polynomial matrices
论文作者
论文摘要
多项式$ n \ times n $矩阵$ a（λ）$和$ b（λ）$在字段$ \ mathbb f $上称为半刻度等效的话，如果存在非字段$ n \ times n $ matrix $ p $，而不是字段$ \ mathbb f $ \ mathbb f $ and times n $ n $ n $ n \ time n $ mathix $ nix $ nix（ f} [λ] $，使得$ a（λ）= p b（λ）q（λ）。$矩阵上矩阵的半标准等价在一个字段$ {\ mathbb f} $中包含两个矩阵家族之间相似性的问题。因此，这些矩阵的等效性可以被认为是线性代数中的困难问题。 本文的目的是在字段$ {\ Mathbb f}上介绍非矩阵$ a（λ）$ a（λ）$ a（λ）$ a（λ）$ {\ mathbb f} $的必要条件。我们还建立了一元多项式矩阵$ a（λ）$和$ b（λ）$的相似性。
Polynomial $n\times n$ matrices $A(λ)$ and $B(λ)$ over a field $\mathbb F $ are called semi-scalar equivalent if there exist a nonsingular $n\times n$ matrix $P$ over the field $\mathbb F $ and an invertible $n\times n$ matrix $Q(λ)$ over the ring ${\mathbb F}[λ]$ such that $A(λ)=P B(λ)Q(λ).$ The semi-scalar equivalence of matrices over a field $ {\mathbb F} $ contain the problem of similarity between two families of matrices. Therefore, these equivalences of matrices can be considered a difficult problem in linear algebra. The aim of the present paper is to present the necessary and sufficient conditions of semi-scalar equivalence of nonsingular matrices $A(λ)$ and $ B(λ) $ over a field ${\mathbb F }$ of characteristic zero in terms of solutions of a homogenous system of linear equations. We also establish similarity of monic polynomial matrices $A(λ)$ and $B(λ)$ over a field.
