\(\S 1\ 行列式因子\)
Def
\( \color{red}{Def} 设\lambda-矩阵A(\lambda)\in M_{m\times n}(\mathbb{F[\lambda]})的秩为r,对于正整数k,1\leq k\leq r,A(\lambda)中必有非零的k级子式.A(\lambda)的全部k级子式的最大公因式D_k(A(\lambda))称为A(\lambda)的\color{red}{k级行列式因子(the\ k-th\ determinant divisor)}. \)
\(例1.\)
\( 设A(\lambda)\in M_n{\mathbb(F[\lambda])},D_1(A(\lambda))=(a_{11}(\lambda),...,a_{1n}(\lambda),a_{21}(\lambda),...,a_{nn}(\lambda)).若rank(A(\lambda))=n,则D_n(A(\lambda))=c\ det(A(\lambda)),其中c\in \mathbb{F}. \)
\(Prop1.\)
\( \color{red}{prop}相抵的\lambda-矩阵有相同的各级行列式因子. \)
\(Thm\)
\( \color{red}{Thm} \begin{pmatrix} d_1{\lambda}\\ &d_2(\lambda)\\ &&...\\ &&&d_r(\lambda)\\ &&&&0\\ &&&&&...\\ &&&&&&0 \end{pmatrix} 为矩阵A(\lambda)的相抵标准型，则d_1(\lambda)=D_1(A(\lambda)),d_2(\lambda)=\frac{D_2(A(\lambda))}{D_1(A(\lambda))},...,d_r(\lambda)=\frac{D_r(A(\lambda))}{D_{r-1}(A(\lambda))}.特别地,\lambda-矩阵A(\lambda)具有唯一的相抵标准型. \)
\(RK\)
\( \color{blue}{RK} 设A,B\in M_n(\mathbb(F)),判断A,B是否相似著需要判断\lambda E_n-A与\lambda E_n-B是否相抵.\\ 有Thm知只需要判断\lambda E_n-A与\lambda E_n-B的相抵标准型是否相同. \)
\(\S2\ 不变因子\)
\(Def\)
\( \color{red}{Def}设]lambda-矩阵A(\lambda)的秩为r.称A(\lambda)的相抵标准型的主对角线上的非零元d_1(\lambda),...,d_r(\lambda)为A(\lambda)的\color{red}{不变因子(invariant factor)}. \)