关于矩阵的秩的专题讨论
$\bf命题：$设$f(x),g(x) \in F[x],A \in {F^{n \times n}}$，且$(f(x),g(x))=1$，则$$r(f(A))+r(g(A))=n+r(f(A)g(A))$$
$\bf命题：$设$A\in P^{n\times m},B\in P^{n\times s},C\in P^{m\times t},D\in P^{s\times t}$，且$r(B)=s$，$AC+BD=0$，则$r\left( {\begin{array}{*{20}{c}}C\\D\end{array}} \right) = t \Longleftrightarrow r\left( C \right) = t$
$\bf命题：$设$n$阶矩阵$A$的特征多项式满足\[f\left( \lambda \right) = g\left( \lambda \right)h\left( \lambda \right),\left( {g\left( \lambda \right),h\left( \lambda \right)} \right) = 1\]证明：$r\left( {g\left( A \right)} \right) = \deg h\left( \lambda \right)，且r\left( {h\left( A \right)} \right) = \deg g\left( \lambda \right)$
$\bf命题：$设$A,B_{i}\in M_{n}(F),rank(A)<n$，且$A=B_{1}B_{2}\cdots B_{k},B_{i}^{2}=B_{i},i=1,2,\cdots,k$，则$$rank(E-A)\leq k(n-rank(A))$$
$\bf命题：$设$A \in {M_{m \times n}}\left( F \right),B \in {M_{n \times s}}\left( F \right)$，且$W = \left\{ {BX \in {F^s}\left| {ABX = 0} \right.} \right\}$，则$\dim W = r\left( B \right) - r\left( {AB} \right)$
$\bf命题：$设$A \in {F^{m \times n}},B \in {F^{p \times n}}$，令$S = \left\{ {Ax|Bx = 0} \right\}$，则$S$作为${F^m}$的子空间的维数为$r\left( {\begin{array}{*{20}{c}}A\\B\end{array}} \right) - r\left( B \right)$
$\bf命题：$设$\sigma \in L\left( V \right)$，$W$为$V$的一个有限维子空间，则\[\dim \sigma \left( W \right) = \dim \left( {Ker\left( \sigma \right) \cap W} \right) = \dim W\]
$\bf命题：$设实二次型$f\left( {{x_1}, \cdots ,{x_n}} \right) = \sum\limits_{i = 1}^n {{{\left( {{a_{i1}}{x_1} + \cdots + {a_{in}}{x_n}} \right)}^2}} $，证明二次型的秩等于$A = {\left( {{a_{ij}}} \right)_{n \times n}}$的秩
$\bf命题：$设$A$为秩为$s$的$n$阶实对称阵，且$x'Ax \ne 0,\forall x \in {R^n}$，记$B = A - {\left( {x'Ax} \right)^{ - 1}}Axx'A$，证明：秩$B=s-1$
$\bf命题：$设$f$为数域$F$上$n$维线性空间$V$上的一个双线性函数，则${L_f}:\alpha \to {\alpha _L}$是$V$到${V^*}$(对偶空间)一个线性映射，其中${\alpha _L}:\beta \to f\left( {\alpha ,\beta } \right),\forall \beta \in V$，则$\dim \left( {{\mathop{\rm Im}\nolimits} {L_f}} \right)=f$的秩($f$的度量矩阵的秩)
$\bf命题：$
附录1
$\bf命题1：$设$A \in {M_{m \times n}}\left( F \right),B \in {M_{n \times p}}\left( F \right)$，则$r\left( {AB} \right) \le \min \left\{ {r\left( A \right),r\left( B \right)} \right\}$
$\bf命题2：$设$A \in {M_{m \times n}}\left( F \right)$，则$r\left( A \right) \le m$且$r\left( A \right) \le n$
$\bf命题3：$$A \in {M_{m \times n}}\left( F \right)$，且$P,Q$为可逆阵，则$r\left( A \right) = r\left( {PA} \right) = r\left( {AQ} \right) = r\left( {PAQ} \right)$
$\bf命题4：$$r\left( {\begin{array}{*{20}{c}}A&0\\0&B\end{array}} \right) = r\left( A \right) + r\left( B \right){\rm{ }},{\rm{ }}r\left( {\begin{array}{*{20}{c}}A&C\\0&B\end{array}} \right) \ge r\left( A \right) + r\left( B \right)$
$\bf命题5：$设$A,B \in {M_{m \times n}}\left( F \right)$，则$r\left( {A + B} \right) \le r\left( A \right) + r\left( B \right)$
$\bf命题6：$设$A \in {M_{m \times n}}\left( F \right),B \in {M_{n \times p}}\left( F \right)$，若$AB = 0$，则$r\left( A \right) + r\left( B \right) \le n$
$\bf命题7：$$r\left( {AB + A + B} \right) \le r\left( A \right) + r\left( B \right)$
$\bf命题8：$设$AB = BA$，则$r\left( {A,B} \right) \le r\left( A \right) + r\left( B \right) - r\left( {AB} \right)$
$\bf命题9：$$\bf（Frobenius公式）$$r\left( {ABC} \right) \ge r\left( {AB} \right) + r\left( {BC} \right) - r\left( B \right)$
$\bf命题10：$$\bf（Sylvester公式）$设$A \in {M_{m \times n}}\left( F \right),B \in {M_{n \times p}}\left( F \right)$，则$r\left( {AB} \right) \geqslant r\left( A \right) + r\left( B \right) - n$
$\bf命题11：$$\bf（秩降阶公式）$设$C \in {M_{m \times n}}\left( F \right),D \in {M_{n \times m}}\left( F \right),m = r\left( A \right) \ge r\left( B \right) = n$，且$A,B$可逆，则\[r\left( {A - D{B^{ - 1}}C} \right) = m - n + r\left( {B - C{A^{ - 1}}D} \right)\]
$\bf命题12：$$\bf（幂等阵与秩）$${A^2} = A \Leftrightarrow r\left( A \right) + r\left( {A - E} \right) = n$
$\bf命题13：$$r\left( {{A^T}A} \right) = r\left( {A{A^T}} \right) = r\left( A \right) = r\left( {{A^T}} \right)$
$\bf命题14：$设$r\left( {AB} \right) = r\left( B \right)$，则$r\left( {ABC} \right) = r\left( {BC} \right)$
$\bf命题15：$设$A$为$n$阶方阵，且$r(A)=r(A^2)$，则对任意的自然数$k$都有$r(A^k)=r(A)$
$\bf命题16：$设$A$为$n$阶可逆阵，则${A^T}{\rm{ = }} - A$当且仅当$r\left( {\begin{array}{*{20}{c}}A&\alpha \\{{\alpha ^T}}&0\end{array}} \right) = r\left( A \right),\forall \alpha \in {F^n}$
$\bf命题17：$设$r\left( A \right) = r\left( {A,b} \right)$，则$r\left( {\begin{array}{*{20}{c}}A&b\\{ - {b^T}}&0\end{array}} \right) = r\left( A \right)$，其中$A$为$n$阶反对称阵，$b$为$n$维列向量
$\bf命题18：$设$A$是数域$F$上的$n$阶方阵，证明：对任意的正整数$k$，有$rank\left( {{A^{n + k}}} \right) = rank\left( {{A^n}} \right)$
