关于子块为矩阵多项式的矩阵的秩

The rank of partitioned matrices with polynomial matrix sub-blocks

为了进一步整合线性代数的内容,利用分块矩阵与λ-多项式理论对子块为矩阵多项式的矩阵的秩进行系统的论述.得到的主要结论:设B(λ)∈F[λ]s×t,A∈F n×n,则rank(B(A))=rank(h1(A))++rank(hr(A)),其中:r=rank(B(λ));h1(λ),,hr(λ)∈F[λ]为任意非零多项式,且h1(λ),,hr(λ)的标准分解式中不可约因子的方幂构成B(λ)的全部初等因子.

In order to further integrate the content of linear algebra,revealed the rank of partitioned λ-matrices,in which every sub-block is matrix polynomial,using partitioned matrix andλ-polynomial theory.The main conclusion is that ifB（λ）∈F [λ ] s ×t,A∈F n ×n,thenrank（B（A））=rank（h1（A））＋ ＋rank（hr（A））.Where r=rank（B（λ））,h1（λ）,,hr（λ） ∈ F [λ] are any non-zero polynomial with the following properties that the powers of irreducible divisors in the canonical form of h1（λ）,,hr（λ） constitute all the elementary divisors ofB（λ）.