线性代数-01矩阵与行列式

矩阵

定义

特殊矩阵：上三角、下三角、对角

操作：初等行列变换、矩阵分块

Jacobi矩阵
A = [ ∂ f 1 ( x 0 ) ∂ x 1 ∂ f 1 ( x 0 ) ∂ x 2 ⋯ ∂ f 1 ( x 0 ) ∂ x n ∂ f 2 ( x 0 ) ∂ x 1 ∂ f 2 ( x 0 ) ∂ x 2 ⋯ ∂ f 2 ( x 0 ) ∂ x n ⋮ ⋮ ⋮ ∂ f m ( x 0 ) ∂ x 1 ∂ f m ( x 0 ) ∂ x 2 ⋯ ∂ f m ( x 0 ) ∂ x n ] A= \begin{bmatrix} \frac{\partial f_1(x_0)}{\partial x_1} & \frac{\partial f_1(x_0)}{\partial x_2} & \cdots & \frac{\partial f_1(x_0)}{\partial x_n}\\ \frac{\partial f_2(x_0)}{\partial x_1} & \frac{\partial f_2(x_0)}{\partial x_2} & \cdots & \frac{\partial f_2(x_0)}{\partial x_n}\\ \vdots & \vdots & & \vdots\\ \frac{\partial f_m(x_0)}{\partial x_1} & \frac{\partial f_m(x_0)}{\partial x_2} & \cdots & \frac{\partial f_m(x_0)}{\partial x_n}\\ \end{bmatrix} A= ∂x1∂f1(x0)∂x1∂f2(x0)⋮∂x1∂fm(x0)∂x2∂f1(x0)∂x2∂f2(x0)⋮∂x2∂fm(x0)⋯⋯⋯∂xn∂f1(x0)∂xn∂f2(x0)⋮∂xn∂fm(x0)

行列式

递归定义：行列式按一行拉普拉斯展开

行列式按一行拉普拉斯分块展开

行列式按k行展开

Jacobi行列式
A = ∣ ∂ f 1 ( x 0 ) ∂ x 1 ∂ f 1 ( x 0 ) ∂ x 2 ⋯ ∂ f 1 ( x 0 ) ∂ x n ∂ f 2 ( x 0 ) ∂ x 1 ∂ f 2 ( x 0 ) ∂ x 2 ⋯ ∂ f 2 ( x 0 ) ∂ x n ⋮ ⋮ ⋮ ∂ f m ( x 0 ) ∂ x 1 ∂ f m ( x 0 ) ∂ x 2 ⋯ ∂ f m ( x 0 ) ∂ x n ∣ A= \begin{vmatrix} \frac{\partial f_1(x_0)}{\partial x_1} & \frac{\partial f_1(x_0)}{\partial x_2} & \cdots & \frac{\partial f_1(x_0)}{\partial x_n}\\ \frac{\partial f_2(x_0)}{\partial x_1} & \frac{\partial f_2(x_0)}{\partial x_2} & \cdots & \frac{\partial f_2(x_0)}{\partial x_n}\\ \vdots & \vdots & & \vdots\\ \frac{\partial f_m(x_0)}{\partial x_1} & \frac{\partial f_m(x_0)}{\partial x_2} & \cdots & \frac{\partial f_m(x_0)}{\partial x_n}\\ \end{vmatrix} A= ∂x1∂f1(x0)∂x1∂f2(x0)⋮∂x1∂fm(x0)∂x2∂f1(x0)∂x2∂f2(x0)⋮∂x2∂fm(x0)⋯⋯⋯∂xn∂f1(x0)∂xn∂f2(x0)⋮∂xn∂fm(x0)

行列式的性质

行列对称： ∣ A T ∣ = ∣ A ∣ \mid A^T\mid=\mid A\mid ∣AT∣=∣A∣

与初等行变换之间的关系:

交换某两行（列）
某两行（列）相同（或倍数）
把某行（列）的倍数加到另一行（列）
…

范德蒙行列式

不等于0的充要条件： a 1 , a 2 , . . . , a n 两两互不相等 a_1,a_2,...,a_n两两互不相等 a1,a2,...,an两两互不相等