本文是“线性代数”系列的第 02 章，主题为“矩阵”。

上一篇：线性代数 Ch01：行列式 · 下一篇：线性代数 Ch03：n维向量

矩阵\(\boldsymbol{A}\)的伴随矩阵
\[ \boldsymbol{A}^*=\left[ \begin{array}{cccc} A_{11} & A_{21} & \cdots & A_{n1}\\ A_{12} & A_{22} & \cdots & A_{n2}\\ \vdots & \vdots & & \vdots\\ A_{1n} & A_{2n} & \cdots & A_{nn}\\ \end{array}\right] \]

2阶矩阵的伴随矩阵: 主对角线元素互换, 副对角线元素变号.

正交矩阵 \(\boldsymbol{A}\boldsymbol{A}^T=\boldsymbol{A}^T\boldsymbol{A}=\boldsymbol{E}\).

\(\boldsymbol{A}\)是正交矩阵\(\Leftrightarrow \boldsymbol{A}^T=\boldsymbol{A}^{-1}\);

\(\boldsymbol{A}\)是正交矩阵\(\Rightarrow|\boldsymbol{A}|^2=1\).

几何意义: 正交矩阵的每个行(列)向量长度均为1, 行(列)向量两两正交.

\(n\)阶矩阵\(\boldsymbol{A}\)可逆

\(\Leftrightarrow|\boldsymbol{A}|\neq0\)

\(\Leftrightarrow r(\boldsymbol{A})=n\)

\(\Leftrightarrow \boldsymbol{A}\)的列(行)向量组线性无关

\(\Leftrightarrow \boldsymbol{A}\)与单位矩阵等价

\(\Leftrightarrow 0\)不是矩阵\(\boldsymbol{A}\)的特征值.

主要公式

(1) 转置

\((k\boldsymbol{A})^T=k\boldsymbol{A}^T\) \((\boldsymbol{A}\boldsymbol{B})^T=\boldsymbol{B}^T\boldsymbol{A}^T\) \((\boldsymbol{A}+\boldsymbol{B})^T=\boldsymbol{A}^T+\boldsymbol{B}^T\)

(2) 可逆

\((k\boldsymbol{A})^{-1}=\dfrac{1}{k}\boldsymbol{A}^{-1}\) \((\boldsymbol{A}\boldsymbol{B})^{-1}=\boldsymbol{B}^{-1}\boldsymbol{A}^{-1}\) \((\boldsymbol{A}^T)^{-1}=(\boldsymbol{A}^{-1})^T\) \((\boldsymbol{A}^n)^{-1}=(\boldsymbol{A}^{-1})^n\)

注意\((\boldsymbol{A}+\boldsymbol{B})^{-1}\)没有运算法则, 利用\(\boldsymbol{E}\)作恒等变形是常用技巧.

(3) 伴随

\((k\boldsymbol{A})^*=k^{n-1}\boldsymbol{A}^*\) \((\boldsymbol{A}\boldsymbol{B})^*=\boldsymbol{B}^*\boldsymbol{A}^*\) \(\boldsymbol{A}\boldsymbol{A}^*=\boldsymbol{A}^*\boldsymbol{A}=|\boldsymbol{A}|\boldsymbol{E}\) \((\boldsymbol{A}^*)^{-1}=(\boldsymbol{A}^{-1})^*=\dfrac{1}{|\boldsymbol{A}|}\boldsymbol{A}\)

\((\boldsymbol{A}^*)^T=(\boldsymbol{A}^T)^*\) \((\boldsymbol{A}^*)^*=|\boldsymbol{A}|^{n-2}\boldsymbol{A}\) \[ r(\boldsymbol{A}^*)=\begin{cases} n & r(\boldsymbol{A})=n\\ 1 & r(\boldsymbol{A})=n-1\\ 0 & r(\boldsymbol{A})<n-1 \end{cases} \]

(4) 秩

\(r(\boldsymbol{A})=r(\boldsymbol{A}^T)=r(\boldsymbol{A}^T\boldsymbol{A})=r(k\boldsymbol{A})\) \(r(\boldsymbol{A}+\boldsymbol{B})\leqslant r(\boldsymbol{A})+r(\boldsymbol{B})\) \(r(\boldsymbol{A}\boldsymbol{B})\leqslant\min(r(\boldsymbol{A}),r(\boldsymbol{B}))\)

\(\boldsymbol{A}\)可逆\(\Rightarrow r(\boldsymbol{A}\boldsymbol{B})=r(\boldsymbol{B}),r(\boldsymbol{B}\boldsymbol{A})=r(\boldsymbol{B})\) \(\boldsymbol{A}\)列满秩\(\Rightarrow r(\boldsymbol{A}\boldsymbol{B})=r(\boldsymbol{B})\)

\(\boldsymbol{A}\)是\(m\times n\)矩阵, \(\boldsymbol{B}\)是\(n\times s\)矩阵, \(\boldsymbol{A}\boldsymbol{B}=\boldsymbol{O}\Rightarrow r(\boldsymbol{A})+r(\boldsymbol{B})\leqslant n\)

\(\boldsymbol{A}\sim \boldsymbol{B}\Rightarrow r(\boldsymbol{A})=r(\boldsymbol{B}),r(\boldsymbol{A}+k\boldsymbol{E})=r(\boldsymbol{B}+k\boldsymbol{E})\) \[ r\left[ \begin{array}{cc} \boldsymbol{A} & \boldsymbol{O}\\ \boldsymbol{O} & \boldsymbol{B}\\ \end{array}\right] =r(\boldsymbol{A})+r(\boldsymbol{B}) \]

(5) 分块矩阵

\[ \left[ \begin{array}{cc} \boldsymbol{A} & \boldsymbol{B}\\ \boldsymbol{C} & \boldsymbol{D}\\ \end{array}\right] ^T=\left[ \begin{array}{cc} \boldsymbol{A}^T & \boldsymbol{C}^T\\ \boldsymbol{B}^T & \boldsymbol{D}^T\\ \end{array}\right] \] \[ \left[ \begin{array}{cc} \boldsymbol{B} & \boldsymbol{O}\\ \boldsymbol{O} & \boldsymbol{C}\\ \end{array}\right] ^n=\left[ \begin{array}{cc} \boldsymbol{B}^n & \boldsymbol{O}\\ \boldsymbol{O} & \boldsymbol{C}^n\\ \end{array}\right] \] \[ \left[ \begin{array}{cc} \boldsymbol{O} & \boldsymbol{B}\\ \boldsymbol{C} & \boldsymbol{O}\\ \end{array}\right] ^{-1}=\left[ \begin{array}{cc} \boldsymbol{O} & \boldsymbol{C}^{-1}\\ \boldsymbol{B}^{-1} & \boldsymbol{O}\\ \end{array}\right] \]

(1) \(\boldsymbol{A}\boldsymbol{B}=\boldsymbol{O}\nRightarrow \boldsymbol{A}=\boldsymbol{O}\)或\(\boldsymbol{B}=\boldsymbol{O}\).

(2) \(\boldsymbol{A}\boldsymbol{B}=\boldsymbol{A}\boldsymbol{C},\boldsymbol{A}\neq \boldsymbol{O}\nRightarrow \boldsymbol{B}=\boldsymbol{C}\).

已知2个\(n\)维列向量\(\boldsymbol{\alpha}\)和\(\boldsymbol{\beta}\)

(1) \(\boldsymbol{\alpha}\boldsymbol{\beta}^T\)和\(\boldsymbol{\beta}\boldsymbol{\alpha}^T\)是\(n\)阶矩阵(互为转置), \(\boldsymbol{\alpha}^T\boldsymbol{\beta}\)和\(\boldsymbol{\beta}^T\boldsymbol{\alpha}\)是一个数(相同), 正好为矩阵\(\boldsymbol{\alpha}\boldsymbol{\beta}^T\)的迹.

(2) \(0\leqslant r(\boldsymbol{\alpha}\boldsymbol{\beta}^T)\leqslant r(\boldsymbol{\alpha})\leqslant1\).

(3) 若\(\boldsymbol{A}=\boldsymbol{\alpha}\boldsymbol{\beta}^T\neq \boldsymbol{O}\Leftrightarrow r(\boldsymbol{A})=1\), 即任两行成比例;

此时有\(\boldsymbol{A}^n=l^{n-1}\boldsymbol{A}\), 其中\(l=\boldsymbol{\alpha}^T\boldsymbol{\beta}=\boldsymbol{\beta}^T\boldsymbol{\alpha}=\displaystyle\sum a_{ii}\).

特殊矩阵的\(n\)次方

(一)

若\[ \boldsymbol{A}=\left[ \begin{array}{ccc} 0 & 0 & 0\\ a & 0 & 0\\ * & b & 0\\ \end{array}\right] \], 则\[ \boldsymbol{A}^2=\left[ \begin{array}{ccc} 0 & 0 & 0\\ 0 & 0 & 0\\ ab & 0 & 0\\ \end{array}\right] ,\boldsymbol{A}^3=\boldsymbol{O} \].

若\[ \boldsymbol{A}=\left[ \begin{array}{cccc} 0 & a & * & *\\ 0 & 0 & b & *\\ 0 & 0 & 0 & c\\ 0 & 0 & 0 & 0\\ \end{array}\right] \], 则\[ \boldsymbol{A}^3=\left[ \begin{array}{cccc} 0 & 0 & 0 & abc\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ \end{array}\right] ,\boldsymbol{A}^4=\boldsymbol{O} \].

(二) 若\(\boldsymbol{P}^{-1}\boldsymbol{A}\boldsymbol{P}=\boldsymbol{B}\), 则\(\boldsymbol{P}^{-1}\boldsymbol{A}^n\boldsymbol{P}=\boldsymbol{B}^n\), 从而\(\boldsymbol{A}^n=\boldsymbol{P}\boldsymbol{B}^n\boldsymbol{P}^{-1}\).

一般\(\boldsymbol{A}\sim\boldsymbol{\Lambda}\)(对角矩阵), \(\boldsymbol{A}^n=\boldsymbol{P}\boldsymbol{\Lambda}^n\boldsymbol{P}^{-1}\), 而对角矩阵的\(n\)次方是容易计算的, 有
\[ \left[ \begin{array}{ccc} \lambda_1 & & \\ & \lambda_2 & \\ & & \lambda_3\\ \end{array}\right] ^n=\left[ \begin{array}{ccc} \lambda_1^n & & \\ & \lambda_2^n & \\ & & \lambda_3^n\\ \end{array}\right] \]

\(\lambda_1,\lambda_2,\lambda_3\)为\(\boldsymbol{A}\)的特征值, \(\boldsymbol{P}=[\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\boldsymbol{\alpha}_3]\), 其中\(\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\boldsymbol{\alpha}_3\)为\(\boldsymbol{A}\)的特征向量.

初等矩阵的\(n\)次幂是同类型的初等矩阵(以初等矩阵的逆即\(n=-1\)为例)

(1) 倍加 \(\boldsymbol{E}_{ij}^{n}(k)=\boldsymbol{E}_{ij}(nk)\)
\[ \left[ \begin{array}{ccc} 1 & & \\ 2 & 1 & \\ & & 1\\ \end{array}\right] ^{-1}=\left[ \begin{array}{ccc} 1 & & \\ -2 & 1 & \\ & & 1\\ \end{array}\right] \]

(2) 互换 \[ \boldsymbol{E}_{ij}^{n}=\begin{cases} \boldsymbol{E} & n=2k\\ \boldsymbol{E}_{ij} & n=2k-1 \end{cases} \]
\[ \left[ \begin{array}{ccc} 1 & & \\ & & 1\\ & 1 & \\ \end{array}\right] ^{-1}=\left[ \begin{array}{ccc} 1 & & \\ & & 1\\ & 1 & \\ \end{array}\right] \]

(3) 乘\(k\) \(\boldsymbol{E}_{i}^{n}(k)=\boldsymbol{E}_{i}(k^n)\)
\[ \left[ \begin{array}{ccc} 1 & & \\ & 1 & \\ & & 2\\ \end{array}\right] ^{-1}=\left[ \begin{array}{ccc} 1 & & \\ & 1 & \\ & & \dfrac{1}{2}\\ \end{array}\right] \]

矩阵经初等变换后秩不变.

补充结论

(1) \[ \left| \begin{array}{cc} \boldsymbol{A} & \boldsymbol{B}\\ \boldsymbol{B} & \boldsymbol{A}\\ \end{array}\right| =|\boldsymbol{A}+\boldsymbol{B}|\cdot|\boldsymbol{A}-\boldsymbol{B}| \]

(2) 设\[ \boldsymbol{H}=\left[ \begin{array}{cc} \boldsymbol{A} & \boldsymbol{C}\\ \boldsymbol{O} & \boldsymbol{B}\\ \end{array}\right] \], 则\[ [\boldsymbol{H},\boldsymbol{E}]=\left[ \begin{array}{cccc} \boldsymbol{A} & \boldsymbol{C} & \boldsymbol{E} & \boldsymbol{O}\\ \boldsymbol{O} & \boldsymbol{B} & \boldsymbol{O} & \boldsymbol{E}\\ \end{array}\right] \xrightarrow{\text{行变换}}\left[ \begin{array}{cccc} \boldsymbol{E} & \boldsymbol{O} & \boldsymbol{A}^{-1} & -\boldsymbol{A}^{-1}\boldsymbol{C}\boldsymbol{B}^{-1}\\ \boldsymbol{O} & \boldsymbol{E} & \boldsymbol{O} & \boldsymbol{B}^{-1}\\ \end{array}\right] \],

\[ \boldsymbol{H}^{-1}=\left[ \begin{array}{cc} \boldsymbol{A}^{-1} & -\boldsymbol{A}^{-1}\boldsymbol{C}\boldsymbol{B}^{-1}\\ \boldsymbol{O} & \boldsymbol{B}^{-1}\\ \end{array}\right] \].

(3) \(\boldsymbol{A}^{-1}\boldsymbol{B}\)可由\([\boldsymbol{A}|\boldsymbol{B}]\xrightarrow{\text{行变换}}[\boldsymbol{E}|\boldsymbol{A}^{-1}\boldsymbol{B}]\)来求.

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