凡人多烦事01

这个屌丝很懒，什么也没留下！

热门标签

行列式与矩阵，运算与性质

作者：凡人多烦事01 | 2024-04-10 02:56:55

踩

行列式与矩阵，运算与性质

行列式计算

一阶行列式：
$\det \left( A \right) = a$

二阶行列式：
$\det \left( A \right) = {a_{11}}{a_{22}} - {a_{12}}{a_{21}}$

三阶行列式：
$\det \left( A \right) = {a_{11}}{a_{22}}{a_{33}} + {a_{12}}{a_{23}}{a_{31}} + {a_{21}}{a_{32}}{a_{13}} - {a_{31}}{a_{22}}{a_{13}} - {a_{21}}{a_{12}}{a_{33}} - {a_{32}}{a_{23}}{a_{11}}$

高阶行列式：利用行列式展开法则求解，行列式等于它的任一行（列）的各元素与其代数余子式的乘积之和。注：代数余子式 ${A_{ij}} = {\left( { - 1} \right)^{i + j}}\det \left( {{M_{ij}}} \right)$ ， $M_{ij}$ 为划掉 $a_{ij}$ 所在行列所得的 $\left( {n - 1} \right)$ 阶方阵。

\begin{array}{l} det (A) = a_{i 1} A_{i 1} + a_{i 2} A_{i 2} + \dots + a_{i n} A_{i n} \\ det (A) = a_{1 j} A_{1 j} + a_{2 j} A_{2 j} + \dots + a_{n j} A_{n j} \end{array}

$\begin{array}{l} \det \left( A \right) = {a_{i1}}{A_{i1}} + {a_{i2}}{A_{i2}} + \cdots + {a_{in}}{A_{in}}\\ \det \left( A \right) = {a_{1j}}{A_{1j}} + {a_{2j}}{A_{2j}} + \cdots + {a_{nj}}{A_{nj}} \end{array}$

det (A) = a_{i 1} A_{i 1} + a_{i 2} A_{i 2} + \dots + a_{in} A_{in} det (A) = a_{1 j} A_{1 j} + a_{2 j} A_{2 j} + \dots + a_{nj} A_{nj}

Matlab求解函数：
$\det \left( A \right)：返回方阵 A 的行列式$

行列式性质

性质1：矩阵与转置矩阵行列式相等。
性质2：互换行列式的两行（列），行列式变号。
性质3：行列式某行（列）的公因子可以提出去。
$\left| {$

\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ ⋮ & ⋮ & ⋮ \\ k a_{i 1} & k a_{i 2} & \dots & k a_{i n} \\ ⋮ & ⋮ & ⋮ \\ a_{n 1} & a_{n 2} & \dots & a_{n n} \end{matrix}

$\begin{array}{c} { {a_{11}}}&{ {a_{12}}}& \cdots &{ {a_{1n}}}\\ \vdots & \vdots &{}& \vdots \\ {k{a_{i1}}}&{k{a_{i2}}}& \cdots &{k{a_{in}}}\\ \vdots & \vdots &{}& \vdots \\ { {a_{n1}}}&{ {a_{n2}}}& \cdots &{ {a_{nn}}} \end{array}$ } \right| = k\left| {

\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ ⋮ & ⋮ & ⋮ \\ a_{i 1} & a_{i 2} & \dots & a_{i n} \\ ⋮ & ⋮ & ⋮ \\ a_{n 1} & a_{n 2} & \dots & a_{n n} \end{matrix}

a_{11} ⋮ k a_{i 1} ⋮ a_{n 1} a_{12} ⋮ k a_{i 2} ⋮ a_{n 2} \dots \dots \dots a_{1 n} ⋮ k a_{in} ⋮ a_{nn} = k a_{11} ⋮ a_{i 1} ⋮ a_{n 1} a_{12} ⋮ a_{i 2} ⋮ a_{n 2} \dots \dots \dots a_{1 n} ⋮ a_{in} ⋮ a_{nn}

性质4：如果行列式的某行（列）是两项之和，那么行列式等于两个行列式之和。
${\left| {$

\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ ⋮ & ⋮ & ⋮ \\ a_{i 1} + a_{i 1}^{'} & a_{i 2} + a_{i 2}^{'} & \dots & a_{i n} + {a^{'}}_{i n} \\ ⋮ & ⋮ & ⋮ \\ a_{n 1} & a_{n 2} & \dots & a_{n n} \end{matrix}

$\begin{array}{c} { {a_{11}}}&{ {a_{12}}}& \cdots &{ {a_{1n}}}\\ \vdots & \vdots &{}& \vdots \\ { {a_{i1}} + a_{i1}^\prime }&{ {a_{i2}} + a_{i2}^\prime }& \cdots &{ {a_{in}} + {a^\prime }_{in}}\\ \vdots & \vdots &{}& \vdots \\ { {a_{n1}}}&{ {a_{n2}}}& \cdots &{ {a_{nn}}} \end{array}$ } \right|} = \left| {

\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ ⋮ & ⋮ & ⋮ \\ a_{i 1} & a_{i 2} & \dots & a_{i n} \\ ⋮ & ⋮ & ⋮ \\ a_{n 1} & a_{n 2} & \dots & a_{n n} \end{matrix}

\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ ⋮ & ⋮ & ⋮ \\ a_{i 1}^{'} & a_{i 2}^{'} & \dots & a_{i n}^{'} \\ ⋮ & ⋮ & ⋮ \\ a_{n 1} & a_{n 2} & \dots & a_{n n} \end{matrix}

$\begin{array}{c} { {a_{11}}}&{ {a_{12}}}& \cdots &{ {a_{1n}}}\\ \vdots & \vdots &{}& \vdots \\ {a_{i1}^\prime }&{a_{i2}^\prime }& \cdots &{a_{in}^\prime }\\ \vdots & \vdots &{}& \vdots \\ { {a_{n1}}}&{ {a_{n2}}}& \cdots &{ {a_{nn}}} \end{array}$ } \right|

a_{11} ⋮ a_{i 1} + a_{i 1}^{'} ⋮ a_{n 1} a_{12} ⋮ a_{i 2} + a_{i 2}^{'} ⋮ a_{n 2} \dots \dots \dots a_{1 n} ⋮ a_{in} + a^{'}_{in} ⋮ a_{nn} = a_{11} ⋮ a_{i 1} ⋮ a_{n 1} a_{12} ⋮ a_{i 2} ⋮ a_{n 2} \dots \dots \dots a_{1 n} ⋮ a_{in} ⋮ a_{nn} + a_{11} ⋮ a_{i 1}^{'} ⋮ a_{n 1} a_{12} ⋮ a_{i 2}^{'} ⋮ a_{n 2} \dots \dots \dots a_{1 n} ⋮ a_{in}^{'} ⋮ a_{nn}

性质5：行列式某行（列）的倍数加到另一行（列），行列式不变。
$\left| {$

\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ ⋮ & ⋮ & ⋮ \\ a_{i 1} & a_{i 2} & \dots & a_{i n} \\ ⋮ & ⋮ & ⋮ \\ a_{j 1} + k a_{i 1} & a_{j 2} + k a_{i 2} & \dots & a_{j n} + k a_{i n} \\ ⋮ & ⋮ & ⋮ \\ a_{n 1} & a_{n 2} & \dots & a_{n n} \end{matrix}

$\begin{array}{c} { {a_{11}}}&{ {a_{12}}}& \cdots &{ {a_{1n}}}\\ \vdots & \vdots &{}& \vdots \\ { {a_{i1}}}&{ {a_{i2}}}& \cdots &{ {a_{in}}}\\ \vdots & \vdots &{}& \vdots \\ { {a_{j1}} + k{a_{i1}}}&{ {a_{j2}} + k{a_{i2}}}& \cdots &{ {a_{jn}} + k{a_{in}}}\\ \vdots & \vdots &{}& \vdots \\ { {a_{n1}}}&{ {a_{n2}}}& \cdots &{ {a_{nn}}} \end{array}$ } \right| = \left| {

\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ ⋮ & ⋮ & ⋮ \\ a_{i 1} & a_{i 2} & \dots & a_{i n} \\ ⋮ & ⋮ & ⋮ \\ a_{j 1} & a_{j 2} & \dots & a_{j n} \\ ⋮ & ⋮ & ⋮ \\ a_{n 1} & a_{n 2} & \dots & a_{n n} \end{matrix}

$\begin{array}{c} { {a_{11}}}&{ {a_{12}}}& \cdots &{ {a_{1n}}}\\ \vdots & \vdots &{}& \vdots \\ { {a_{i1}}}&{ {a_{i2}}}& \cdots &{ {a_{in}}}\\ \vdots & \vdots &{}& \vdots \\ { {a_{j1}}}&{ {a_{j2}}}& \cdots &{ {a_{jn}}}\\ \vdots & \vdots &{}& \vdots \\ { {a_{n1}}}&{ {a_{n2}}}& \cdots &{ {a_{nn}}} \end{array}$ } \right|

a_{11} ⋮ a_{i 1} ⋮ a_{j 1} + k a_{i 1} ⋮ a_{n 1} a_{12} ⋮ a_{i 2} ⋮ a_{j 2} + k a_{i 2} ⋮ a_{n 2} \dots \dots \dots \dots a_{1 n} ⋮ a_{in} ⋮ a_{jn} + k a_{in} ⋮ a_{nn} = a_{11} ⋮ a_{i 1} ⋮ a_{j 1} ⋮ a_{n 1} a_{12} ⋮ a_{i 2} ⋮ a_{j 2} ⋮ a_{n 2} \dots \dots \dots \dots a_{1 n} ⋮ a_{in} ⋮ a_{jn} ⋮ a_{nn}

矩阵性质

加法：

\begin{array}{l} A + B = B + A \\ (A + B) + C = A + (B + C) \end{array}

$\begin{array}{l} A + B = B + A\\ \left( {A + B} \right) + C = A + \left( {B + C} \right) \end{array}$

A + B = B + A (A + B) + C = A + (B + C)

数乘：

\begin{array}{l} (α β) A = α (β A) \\ (α + β) A = α A + β A \end{array}

$\begin{array}{l} \left( {\alpha \beta } \right)A = \alpha \left( {\beta A} \right)\\ (\alpha + \beta )A = \alpha A + \beta A \end{array}$

(α β) A = α (β A) (α + β) A = α A + β A

乘法：满足结合律，不满足交换律。

\begin{array}{l} A B \neq B A \\ (A B) C = A (B C) \\ A (B + C) = A B + A C \\ (A + B) C = A C + B C \\ α (A B) = (α A) B = A (α B) \end{array}

$\begin{array}{l} AB \ne BA\\ \left( {AB} \right)C = A\left( {BC} \right)\\ A\left( {B + C} \right) = AB + AC\\ \left( {A + B} \right)C = AC + BC\\ \alpha (AB) = (\alpha A)B = A(\alpha B) \end{array}$

A B \neq = B A (A B) C = A (BC) A (B + C) = A B + A C (A + B) C = A C + BC α (A B) = (α A) B = A (α B)

行列式：
${A B}|=|{A}||{B}|$

转置：矩阵 $A$ 的转置，记作 ${A^T}$

\begin{array}{l} {(A^{T})}^{T} = A \\ {(A + B)}^{T} = A^{T} + B^{T} \\ {(A B)}^{T} = B^{T} A^{T} \\ | A^{T} | = | A | \end{array}

$\begin{array}{l} {\left( { {A^T}} \right)^T} = A\\ {\left( {A + B} \right)^T} = {A^T} + {B^T}\\ {\left( {AB} \right)^T} = {B^T}{A^T}\\ \left| { {A^T}} \right| = \left| A \right| \end{array}$

(A^{T})^{T} = A (A + B)^{T} = A^{T} + B^{T} (A B)^{T} = B^{T} A^{T} A^{T} = ∣ A ∣

共轭转置：矩阵 $A$ 的共轭转置，记作 ${A^H}$

\begin{array}{l} {(A^{H})}^{H} = A \\ {(A + B)}^{H} = A^{H} + B^{H} \\ {(A B)}^{H} = B^{H} A^{H} \\ | A^{H} | = {| A |}^{H} \\ {(k A)}^{H} = k^{H} A^{H} \end{array}

$\begin{array}{l} {\left( { {A^H}} \right)^H} = A\\ {\left( {A + B} \right)^H} = {A^H} + {B^H}\\ {\left( {AB} \right)^H} = {B^H}{A^H}\\ \left| { {A^H}} \right| = {\left| A \right|^H}\\ {\left( {kA} \right)^H} = {k^H}{A^H} \end{array}$

(A^{H})^{H} = A (A + B)^{H} = A^{H} + B^{H} (A B)^{H} = B^{H} A^{H} A^{H} = ∣ A ∣^{H} (k A)^{H} = k^{H} A^{H}

逆性质：

\begin{array}{l} {(A B)}^{- 1} = B^{- 1} A^{- 1} \\ {(A^{T})}^{- 1} = {(A^{- 1})}^{T} \\ A 可 逆 \Leftrightarrow A 满 秩 \Leftrightarrow | A | \neq 0 \Leftrightarrow A 为 非 奇 异 矩 阵 \end{array}

$\begin{array}{l} {\left( {AB} \right)^{ - 1}} = {B^{ - 1}}{A^{ - 1}}\\ {\left( { {A^T}} \right)^{ - 1}} = {\left( { {A^{ - 1}}} \right)^T}\\ A可逆 \Leftrightarrow A满秩 \Leftrightarrow \left| A \right| \ne 0 \Leftrightarrow A为非奇异矩阵 \end{array}$

(A B)^{- 1} = B^{- 1} A^{- 1} (A^{T})^{- 1} = (A^{- 1})^{T} A 可逆 \Leftrightarrow A 满秩 \Leftrightarrow ∣ A ∣ \neq = 0 \Leftrightarrow A 为非奇异矩阵

偏导数1：标量对一维矩阵求偏导， $X$ 为 $\left( {n \times 1} \right)$ 向量。

\begin{array}{l} \frac{\partial A_{1 \times n} X}{X} = A^{T} \\ \frac{\partial X^{T} A_{n \times 1}}{X} = A \\ \frac{\partial X^{T} A_{n \times n} X}{X} = A X + A^{T} X \\ \frac{\partial X^{T} A^{T} A X}{X} = 2 A^{T} A X \end{array}

$\begin{array}{l} \frac{ {\partial {A_{1 \times n}}X}}{X} = {A^T}\\ \frac{ {\partial {X^T}{A_{n \times 1}}}}{X} = A\\ \frac{ {\partial {X^T}{A_{n \times n}}X}}{X} = AX + {A^T}X\\ \frac{ {\partial {X^T}{A^T}AX}}{X} = 2{A^T}AX \end{array}$

\frac{\partial A _{1 \times n} X}{X} = A^{T} \frac{\partial X ^{T} A _{n \times 1}}{X} = A \frac{\partial X ^{T} A _{n \times n} X}{X} = A X + A^{T} X \frac{\partial X ^{T} A ^{T} A X}{X} = 2 A^{T} A X

偏导数2：标量对二维矩阵求偏导， $X$ 为 $\left( {m \times n} \right)$ 矩阵。

\begin{array}{l} \frac{\partial A_{1 \times m} X B_{n \times 1}}{X} = A^{T} B^{T} \Leftrightarrow \frac{\partial A^{T} X B}{X} = A B^{T} \\ \frac{\partial A_{1 \times n} X^{T} B_{m \times 1}}{X} = B A \Leftrightarrow \frac{\partial A^{T} X^{T} B}{X} = B A^{T} \end{array}

$\begin{array}{l} \frac{ {\partial {A_{1 \times m}}X{B_{n \times 1}}}}{X} = {A^T}{B^T} \Leftrightarrow \frac{ {\partial {A^T}XB}}{X} = A{B^T}\\ \frac{ {\partial {A_{1 \times n}}{X^T}{B_{m \times 1}}}}{X} = BA \Leftrightarrow \frac{ {\partial {A^T}{X^T}B}}{X} = B{A^T} \end{array}$

\frac{\partial A _{1 \times m} X B _{n \times 1}}{X} = A^{T} B^{T} \Leftrightarrow \frac{\partial A ^{T} XB}{X} = A B^{T} \frac{\partial A _{1 \times n} X ^{T} B _{m \times 1}}{X} = B A \Leftrightarrow \frac{\partial A ^{T} X ^{T} B}{X} = B A^{T}

参考文献

参考1：线性代数与空间解析几何（郑宝东）

声明：本文内容由网友自发贡献，不代表【wpsshop博客】立场，版权归原作者所有，本站不承担相应法律责任。如您发现有侵权的内容，请联系我们。转载请注明出处：https://www.wpsshop.cn/w/凡人多烦事01/article/detail/396231