矩阵

概念

1. 正交矩阵：
   $$A^T=A^{-1}$$
   $$|A{|}^2=1$$
2. 可逆矩阵：
   $$|A|\ne 0$$
   $$r(A)=n$$
   $$A的列向量组线性无关$$
   $$A=P_1{P}_2\cdots P_i$$
   $$A与单位矩阵等价$$
   $$0不是特征值$$

公式

1. 可逆
   $$(A^n{)}^{-1}=(A^{-1}{)}^n$$
   $$(A^T{)}^{-1}=(A^{-1}{)}^T$$
   $$|A^{-1}|=|A{|}^{-1}=\frac{1}{|A|}{A}_{-1}=\frac{1}{|A|}{A}^{\ast }$$
2. 伴随
   $$|A^{\ast }|=|A{|}^{n-1}$$
   $${\left(A^{\ast }\right)}^{-1}={\left(A^{-1}\right)}^{\ast }=\frac{1}{|A|}A$$
   $${\left(A^{\ast }\right)}^T={\left(A^T\right)}^{\ast }$$
   $${\left(kA\right)}^{\ast }=k^{n-1}{A}^{\ast }$$
   $${\left(A^{\ast }\right)}^{\ast }=|A{|}^{n-2}A$$
   $$r\left(A^{\ast }\right)=\{\begin{array}{cc}n,& r(A)=n\\ 1,& r(A)=n-1\\ 0,& r(A)<n-1\end{array}$$
3. 秩
   $$r(A)=r(A^T)$$
   $$r(A^{T}A)=r(A)$$
   $$当k\ne 0时：r(kA)=r(A)$$
   $$r(A+B)\le r(A)+r(B)$$
   $$r(AB)\le \min (r(A),r(B))$$
   $$若A是m\times n矩阵,B是n\times s矩阵,AB=O,则：r(A)+r(B)\le n$$
   $$r\left[\begin{array}{cc}A& O\\ O& B\end{array}\right]=r(A)+r(B)$$
   $$若A\sim B,则r(A)=r(B),r(A+kE)=r(B+kE)$$
   $$AB等价\iff r(A)=r(A|B)=r(B)\iff 互相表出$$
4. 分块矩阵
   $${\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]}^T=\left[\begin{array}{cc}{A}^T& C^T\\ B^T& D^T\end{array}\right]$$
   $${\left[\begin{array}{cc}B& O\\ O& C\end{array}\right]}^{-1}=\left[\begin{array}{cc}{B}^{-1}& O\\ O& C^{-1}\end{array}\right]$$
   $${\left[\begin{array}{cc}O& B\\ C& O\end{array}\right]}^{-1}=\left[\begin{array}{cc}O& C^{-1}\\ B^{-1}& O\end{array}\right]$$
   $$A=\left[\begin{array}{ccc}0& a& b\\ 0& 0& c\\ 0& 0& 0\end{array}\right],则A^2=\left[\begin{array}{ccc}0& 0& ac\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$$
   $$A=\left[\begin{array}{cccc}0& d& a& b\\ 0& 0& e& c\\ 0& 0& 0& f\\ 0& 0& 0& 0\end{array}\right],则A^3=\left[\begin{array}{cccc}0& 0& 0& def\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$$

运算法则

1. $AB=O$推不出$A=O,B=O$
2. $AB=AC,A\ne C$推不出$B=C$
3. $|AB|=|A|\cdot |B|$