7.1. 常用总结

7.1.1. 顺序主子式

设有 \(n\) 阶方阵 \({\bm A}\)

\[{\bm A} = \left[ {\begin{array}{cccc} {{a_{11}}}&{{a_{12}}}& \cdots &{{a_{1n}}}\\ {{a_{21}}}&{{a_{22}}}& \cdots &{{a_{2n}}}\\ \vdots & \vdots & \ddots & \vdots \\ {{a_{n1}}}&{{a_{n2}}}& \cdots &{{a_{nn}}} \end{array}} \right] \]

则其 \(i, (0 \leq i \leq n)\) 阶顺序主子式为行列式

\[D_i = \left| {\begin{array}{cccc} {{a_{11}}}&{{a_{12}}}& \cdots &{{a_{1i}}}\\ {{a_{21}}}&{{a_{22}}}& \cdots &{{a_{2i}}}\\ \vdots & \vdots & \ddots & \vdots \\ {{a_{i1}}}&{{a_{i2}}}& \cdots &{{a_{ii}}} \end{array}} \right| \]

矩阵 \({\bm A}\) 所有阶顺序主子式 \(D_1, D_2, \cdots, D_n\) 组成 \({\bm A}\) 的顺序主子式.

  • 矩阵 \({\bm A}\) 为正定矩阵的充要条件是 \({\bm A}\) 的所有顺序主子式 \(D_i\) 大于零;

  • 矩阵 \({\bm A}\) 有唯一LU分解的充要条件是 \({\bm A}\) 的所有顺序主子式 \(D_i\) 不等于零;

7.1.2. 伴随矩阵

定义 : 对于方阵 \({\bm A} = (a_{ij})_{n\times n}\) , 将矩阵 \({\bm A}\) 的第 \(i\) 行第 \(j\) 列去掉后, 剩下的元素按原来的顺序组成一个新的 \(n-1\) 阶矩阵, 新矩阵的行列式称为元素 \(a_{ij}\) 的 余子式 , 记为 \(M_{ij}\) ; 称 \(A_{ij} = (-1)^{i+j}M_{ij}\) 为 \(a_{ij}\) 的代数余子式; 将所有元素的代数余子式按如下规律组成一个矩阵, 将此矩阵称为方阵 \({\bm A}\) 的 伴随矩阵 , 记为: 方阵 \({\bm A}^{*}\)

\[{\bm A}^{*} = \left[ {\begin{array}{cccc} {{A_{11}}}&{{A_{21}}}& \cdots &{{A_{n1}}}\\ {{A_{12}}}&{{A_{22}}}& \cdots &{{A_{n2}}}\\ \vdots & \vdots & \vdots & \vdots \\ {{A_{1n}}}&{{A_{2n}}}& \ldots &{{A_{nn}}} \end{array}} \right] \]

性质:

  • \({\bm A}\) 可逆 \(\leftrightarrow\) \({\bm A}^*\) 可逆

  • \({\bm A}\) 可逆 \(\leftrightarrow\) \(({\bm A}^{-1})^* = ({\bm A}^*)^{-1}\)

  • \({\bm A}\) 可逆 \(\leftrightarrow\) \({\bm A}^* = |{\bm A}|{\bm A}^{-1}\)

  • \(({\bm A}^*)^* = (|{\bm A}|{\bm A}^{-1})^* = |{\bm A}|^{n-1}({\bm A}^*)^{-1}\)

  • \(({\bm A}^T)^* = ({\bm A}^*)^T\)

  • \((k{\bm A})^* = k^{n-1}{\bm A}^*\)

  • \(({\bm A}{\bm B})^* = {\bm B}^*{\bm A}^*\)

  • \(|{\bm A}^*| = |{\bm A}|^{n-1}\)

7.1.3. 矩阵的逆

性质

设 \({\bm A}, {\bm B}\) 为 \(n\) 阶可逆矩阵, 数 \(\lambda \neq 0\) , 则

  • \(({\bm A}^{-1})^{-1} = {\bm A}\)

  • \(({\bm A}^T)^{-1} = ({\bm A}^{-1})^T\)

  • \((\lambda{\bm A})^{-1} = \lambda^{-1}{\bm A}^{-1}\)

  • \(({\bm A}{\bm B})^{-1} = {\bm B}^{-1}{\bm A}^{-1}\)

特殊矩阵的逆

二阶方阵的逆

\[{\left[ {\begin{array}{ccc}a&b\\c&d\end{array}} \right]^{ - 1}} = \frac{1}{{ad - bc}}\left[ {\begin{array}{ccc}d&{ - b}\\{ - c}&a\end{array}} \right] \]

提示

上述公式源于伴随矩阵求逆原理.

三角矩阵的逆

\[{\bm{C}} = \left[ {\begin{array}{ccccc}1&{}&{}&{}&{}\\1&1&{}&{}&{}\\1&1& \ddots &{}&{}\\ \vdots & \vdots & \ddots &1&{}\\1&1& \cdots &1&1\end{array}} \right],\;{{\bm{C}}^{ - 1}} = \left[ {\begin{array}{ccccc}1&{}&{}&{}&{}\\{ - 1}&1&{}&{}&{}\\{}&{ - 1}& \ddots &{}&{}\\{}&{}& \ddots &1&{}\\{}&{}&{}&{ - 1}&1\end{array}} \right] \]

求解方法

以以下矩阵为例:

\({\bm A} = \left[ {\begin{array}{ccc}1&2&2\\2&1&2\\2&0&1\end{array}} \right]\)

伴随矩阵求逆

由 \({\bm A}^* = |{\bm A}|{\bm A}^{-1}\) 知 \({\bm A}^{-1} = \frac{1}{|{\bm A}|}{\bm A}^*\) , 故可根据此式计算.

例如:

\[\begin{array}{l} {A_{11}} = \;\,{( - 1)^{1 + 1}}\left| {\begin{array}{ccc} 1&2\\ 0&1 \end{array}} \right| = 1,\;\,\,\,\,{A_{12}} = {( - 1)^{1 + 2}}\left| {\begin{array}{ccc} 2&2\\ 2&1 \end{array}} \right| = 2,\;\,\,\,\,{A_{13}} = {( - 1)^{1 + 3}}\left| {\begin{array}{ccc} 2&1\\ 2&0 \end{array}} \right| = - 2\\ {A_{21}} = {( - 1)^{2 + 1}}\left| {\begin{array}{ccc} 2&2\\ 0&1 \end{array}} \right| = - 2,\;{A_{22}} = {( - 1)^{2 + 2}}\left| {\begin{array}{ccc} 1&2\\ 2&1 \end{array}} \right| = - 3,\;{A_{23}} = {( - 1)^{2 + 3}}\left| {\begin{array}{ccc} 1&2\\ 2&0 \end{array}} \right| = 4\\ {A_{31}} = {( - 1)^{3 + 1}}\left| {\begin{array}{ccc} 2&2\\ 1&2 \end{array}} \right| = 2,\;\;\;{A_{32}} = {( - 1)^{3 + 2}}\left| {\begin{array}{ccc} 1&2\\ 2&2 \end{array}} \right| = 2,\;{A_{33}} = {( - 1)^{3 + 3}}\left| {\begin{array}{ccc} 1&2\\ 2&1 \end{array}} \right| = - 3 \end{array} \]

从而有

\[{{\bm{A}}^{1}} = \frac{1}{|{\bm A}|}\left[ {\begin{array}{ccc} {{A_{11}}}&{{A_{21}}}&{{A_{31}}}\\ {{A_{12}}}&{{A_{22}}}&{{A_{32}}}\\ {{A_{13}}}&{{A_{23}}}&{{A_{33}}} \end{array}} \right] = \left[ {\begin{array}{ccc} 1&{ - 2}&2\\ 2&{ - 3}&{ 2}\\ { - 2}&{ 4}&{ - 3} \end{array}} \right] \]

初等行变换求逆

将待求逆矩阵与单位矩阵拼成一个矩阵, 对新矩阵只进行 初等行变换 , 使得待求逆矩阵部分变为单位矩阵, 那么对应的原始的单位阵变为待求逆矩阵的逆, 即

\[[{\bm A} | {\bm I}] \rightarrow [{\bm I} | {\bm A}^{-1}] \]

例如:

some text

图 7.1 初等变换求逆

初等变换求逆

Sherman-Morrison-Woodbury公式

设 \({\bm A}\in {\mathbb R}^{n\times n}\) 非奇异, \({\bm u}, {\bm v}\in {\mathbb R}^n\) , 则有 Sherman-Morrison 等式

(7.1)\[\left({\bm A} + {\bm u} {\bm v}^{T}\right)^{-1}={\bm A}^{-1}-{\bm A}^{-1} {\bm u}\left({\bm I} + {\bm v}^{T} {\bm A}^{-1} {\bm u}\right)^{-1} {\bm v}^{T} {\bm A}^{-1}. \]

提示

Sherman-Morrison等式可以通过求解线性方程组 \(({\bm A} + {\bm u}{\bm v}^T){\bm x} = {\bm b}\) 得到.

由 Sherman-Morrison 公 式.7.1 , 令 \({\bm U}\in {\mathbb R}^{n\times k}\) , \({\bm V}\in {\mathbb R}^{n\times k}\) , 则有 Sherman-Morrison-Woodbury 等式

(7.2)\[\left({\bm A} + {\bm U} {\bm V}^{T}\right)^{-1}={\bm A}^{-1}-{\bm A}^{-1} {\bm U}\left({\bm I} + {\bm V}^{T} {\bm A}^{-1} {\bm U}\right)^{-1} {\bm V}^{T} {\bm A}^{-1}. \]

7.1.4. 矩阵的秩

性质

  • \(0 \leq {\rm rank}({\bm A}_{m \times n}) \leq {\rm min} \{m, n\}\)

  • \({\rm rank}({\bm A}^T) = {\rm rank}({\bm A})\)

  • \({\rm rank}({\bm A}{\bm B}) \leq {\rm min}\{ {\rm rank}({\bm A}), {\rm rank}({\bm B})\}\)

  • \({\rm rank}({\bm A}+{\bm B}) \leq {\rm rank}({\bm A}) + {\rm rank}({\bm B})\)

  • \({\rm max}\{ {\rm rank}({\bm A}), {\rm rank}({\bm B})\} \leq {\rm rank}({\bm A}, {\bm B}) \leq {\rm rank}({\bm A}) + {\rm rank}({\bm B})\)

  • 若 \({\bm A} \sim {\bm B}\) , 则 \({\rm rank}({\bm A}) = {\rm rank}({\bm B})\)

  • 若 \({\bm P} , {\bm Q}\) 可逆 , 则 \({\rm rank}({\bm P}{\bm A}{\bm Q}) = {\rm rank}({\bm A})\)

  • 若 \({\bm A}_{m\times n}{\bm B}_{n\times l} = {\bm O}\) , 则 \({\rm rank}({\bm A}) + {\rm rank}({\bm B}) \leq n\)

  • 若 \({\bm A}\) 为列满秩矩阵, 且 \({\bm A}{\bm B} = {\bm O}\) , 则 \({\bm B} = {\bm O}\)

7.1.5. 矩阵的迹行列式特征值

  • 方阵 \({\bm A}\) 的行列式与特征值的关系: \(|{\bm A}| = \lambda_1\lambda_2 \cdots \lambda_n\)

  • 方阵 \({\bm A}\) 的迹与特征值的关系: \({tr}({\bm A}) = \lambda_1 + \lambda_2 + \cdots + \lambda_n\)