矩阵及其运算

一､矩阵概念

$m\ast n$的矩阵

$$A=\left[\begin{array}{cccc}{a}_{11}& a_{12}& ...& a_{1n}\\ a_{21}& a_{22}& ...& a_{2n}\\ ...& ...& ...& ...\\ a_{m1}& a_{m2}& ...& a_{mn}\end{array}\right]$$
当m=n时，为n阶矩阵，亦称为n阶方阵。

行矩阵：$1\ast n$的矩阵；

$$\left[\begin{array}{cccc}{a}_{11}& a_{12}& ...& a_{1n}\end{array}\right]$$

列矩阵：$m\ast 1$的矩阵；

$$\left[\begin{array}{c}{a}_{11}\\ a_{21}\\ ...\\ a_{m1}\end{array}\right]$$

零矩阵：元素都为0的矩阵，记作$O$；

对角矩阵：除对角线外的元素都为0的矩阵，记作$\Lambda$；

$$\Lambda =\left[\begin{array}{cccc}{\lambda }_1& 0& ...& 0\\ 0& {\lambda }_2& ...& 0\\ ...& ...& ...& ...\\ 0& 0& ...& {\lambda }_n\end{array}\right]$$
记作：$\Lambda =diag({\lambda }_1,{\lambda }_2,...,{\lambda }_n)$

单位矩阵：对角线为1,其余元素都为0的矩阵；

$$E=\left[\begin{array}{cccc}1& 0& ...& 0\\ 0& 1& ...& 0\\ ...& ...& ...& ...\\ 0& 0& ...& 1\end{array}\right]$$

二､矩阵运算

已知$A=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]$,$B=\left[\begin{array}{cc}{b}_{11}& b_{12}\\ b_{21}& b_{22}\end{array}\right]$,常数$\lambda$;

1､矩阵相加

$A+B=\left[\begin{array}{cc}{a}_{11}+b_{11}& a_{12}+b_{12}\\ a_{21}+b_{21}& a_{22}+b_{22}\end{array}\right]$

2､数与矩阵相乘

$\lambda A=\left[\begin{array}{cc}\lambda a_{11}& \lambda a_{12}\\ \lambda a_{21}& \lambda a_{22}\end{array}\right]$

3､矩阵相乘

$AB=\left[\begin{array}{cc}{a}_{11}{b}_{11}+a_{12}{b}_{21}& a_{11}{b}_{12}+a_{12}{b}_{22}\\ a_{21}{b}_{11}+a_{22}{b}_{21}& a_{21}{b}_{12}+a_{22}{b}_{22}\end{array}\right]$

三、矩阵自身的运算

1､转置矩阵

$A=\left[\begin{array}{cccc}{a}_{11}& a_{12}& ...& a_{1n}\\ a_{21}& a_{22}& ...& a_{2n}\\ ...& ...& ...& ...\\ a_{m1}& a_{m2}& ...& a_{mn}\end{array}\right]，A^T=\left[\begin{array}{cccc}{a}_{11}& a_{21}& ...& a_{m1}\\ a_{12}& a_{22}& ...& a_{m2}\\ ...& ...& ...& ...\\ a_{1n}& a_{2n}& ...& a_{mn}\end{array}\right]$

转置矩阵的运算公式：
$(A^T{)}^T=A$
$(A+B{)}^T=A^T+B^T$
$(\lambda A{)}^T=\lambda A^T$
$(AB{)}^T=B^T{A}^T$

2､方阵的行列式

运算公式：
$\left|A^T\right|=\left|A\right|$
$\left|\lambda A\right|={\lambda }^n\left|A\right|$
$\left|AB\right|=\left|A\right|\left|B\right|$

伴随矩阵：由各元素的代数余子式所组成的矩阵；

$$A^{\ast }=\left[\begin{array}{cccc}{A}_{11}& A_{21}& ...& A_{n1}\\ A_{12}& A_{22}& ...& A_{n2}\\ ...& ...& ...& ...\\ A_{1n}& A_{2n}& ...& A_{nn}\end{array}\right]$$

$A{A}^{\ast }=A^{\ast }A=\left|A\right|E$

3､逆矩阵

$AB=BA=E$,则矩阵$B$称为$A$的逆矩阵,$|A|̸=0$。

$$A^{-1}=\frac{1}{|A|}{A}^{\ast }$$

运算公式：
如果$A=P\Lambda P^{-1}$,$\Lambda =diag({\lambda }_1,{\lambda }_2,...,{\lambda }_n)$那么
$$A^k=P{\Lambda }^k{P}^{-1}$$
$$\phi (A)=P\phi (\Lambda )P^{-1}$$
φ(Λ)=diag{φ(λ1),φ(λ2),...,φ(λn)}\varphi(\Lambda)=diag\{\varphi(\lambda_1),\varphi(\lambda_2),...,\varphi(\lambda_n) \}φ(Λ)=diag{φ(λ1​),φ(λ2​),...,φ(λn​)}

四、克拉默法则

如果线性方程组的系数矩阵$A$的行列式不等于0,那么方程组有唯一解：
$$x_i=\frac{|A_i|}{|A|}$$
其中$A_i$是把$A$中的第$i$列使用常数项矩阵代替后所得到的n阶矩阵，即
$$A_i=\left[\begin{array}{cccccc}{a}_{11}& a_{12}& ...& b_1& ...& a_{1n}\\ a_{21}& a_{22}& ...& b_2& ...& a_{2n}\\ ...& ...& ...& ...& ...& ...\\ a_{m1}& a_{m2}& ...& b_n& ...& a_{mn}\end{array}\right]$$

五、矩阵分块法

$A=\left[\begin{array}{ccc}{A}_{11}& ...& A_{1r}\\ ...& ...& ...\\ A_{s1}& ...& A_{sr}\end{array}\right],B=\left[\begin{array}{ccc}{B}_{11}& ...& B_{1r}\\ ...& ...& ...\\ B_{s1}& ...& B_{sr}\end{array}\right]$

运算法则：

1､加法

$A+B=\left[\begin{array}{ccc}{A}_{11}+B_{11}& ...& A_{1r}+B_{1r}\\ ...& ...& ...\\ A_{s1}+B_{s1}& ...& A_{sr}+B_{sr}\end{array}\right]$

2､与常数相乘

$\lambda A=\left[\begin{array}{ccc}\lambda A_{11}& ...& \lambda A_{1r}\\ ...& ...& ...\\ \lambda A_{s1}& ...& \lambda A_{sr}\end{array}\right]$

3､矩阵相乘

$AB=\left[\begin{array}{ccc}{C}_{11}& ...& C_{1r}\\ ...& ...& ...\\ C_{s1}& ...& C_{sr}\end{array}\right]$

其中

$C_{ij}={\sum }_{k=1}^t{A}_{ik}{B}_{kj}$

4､转置

$A^T=\left[\begin{array}{ccc}{A}_{11}^T& ...& A_{s1}^t\\ ...& ...& ...\\ A_{1r}^T& ...& A_{sr}^T\end{array}\right]$

5､分块对角矩阵

$A=\left[\begin{array}{cccc}{A}_1& & & O\\ & A_2& & \\ & & ...& \\ O& & & A_s\end{array}\right]$

则有：
$|A|=|A_1||A_2|...|A_s|$

$A^{-1}=\left[\begin{array}{cccc}{A}_1^{-1}& & & O\\ & A_2^{-1}& & \\ & & ...& \\ O& & & A_s^{-1}\end{array}\right]$