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# 海森矩阵

## 海森矩阵

$\mathbf{\text{H}}f=\left[\begin{array}{cccc}\frac{{\partial }^{2}f}{\partial {x}^{2}}& \frac{{\partial }^{2}f}{\partial x\partial y}& \frac{{\partial }^{2}f}{\partial x\partial z}& \cdots \\ \\ \frac{{\partial }^{2}f}{\partial y\partial x}& \frac{{\partial }^{2}f}{\partial {y}^{2}}& \frac{{\partial }^{2}f}{\partial y\partial z}& \cdots \\ \\ \frac{{\partial }^{2}f}{\partial z\partial x}& \frac{{\partial }^{2}f}{\partial z\partial y}& \frac{{\partial }^{2}f}{\partial {z}^{2}}& \cdots \\ \\ ⋮& ⋮& ⋮& \ddots \end{array}\right]$

• 这只对标量值函数有意义。
• 此对象$\mathbf{\text{H}}f$不是普通矩阵;它是一个矩阵与 函数 作为元素。 换句话说，它的目的是在某个点计算 $\left({x}_{0},{y}_{0},\dots \right)$
$\mathbf{\text{H}}f\left({x}_{0},{y}_{0},\dots \right)=\left[\begin{array}{ccc}\frac{{\partial }^{2}f}{\partial {x}^{2}}\left({x}_{0},{y}_{0},\dots \right)& \frac{{\partial }^{2}f}{\partial x\partial y}\left({x}_{0},{y}_{0},\dots \right)& \cdots \\ \\ \frac{{\partial }^{2}f}{\partial y\partial x}\left({x}_{0},{y}_{0},\dots \right)& \frac{{\partial }^{2}f}{\partial {y}^{2}}\left({x}_{0},{y}_{0},\dots \right)& \cdots \\ \\ ⋮& ⋮& \ddots \end{array}\right]$

## 示例: 计算海森

$\begin{array}{rl}\phantom{\rule{1em}{0ex}}{f}_{x}\left(x,y\right)& =\frac{\partial }{\partial x}\left({x}^{3}-2xy-{y}^{6}\right)=3{x}^{2}-2y\\ \\ {f}_{y}\left(x,y\right)& =\frac{\partial }{\partial y}\left({x}^{3}-2xy-{y}^{6}\right)=-2x-6{y}^{5}\end{array}$

$\begin{array}{rl}{f}_{xx}\left(x,y\right)& =\frac{\partial }{\partial x}\left(3{x}^{2}-2y\right)=6x\\ \\ {f}_{xy}\left(x,y\right)& =\frac{\partial }{\partial y}\left(3{x}^{2}-2y\right)=-2\\ \\ {f}_{yx}\left(x,y\right)& =\frac{\partial }{\partial x}\left(-2x-6{y}^{5}\right)=-2\\ \\ {f}_{yy}\left(x,y\right)& =\frac{\partial }{\partial y}\left(-2x-6{y}^{5}\right)=-30{y}^{4}\end{array}$

$\mathbf{\text{H}}f\left(x,y\right)=\left[\begin{array}{cc}{f}_{xx}\left(x,y\right)& {f}_{yx}\left(x,y\right)\\ {f}_{xy}\left(x,y\right)& {f}_{yy}\left(x,y\right)\end{array}\right]=\left[\begin{array}{cc}6x& -2\\ -2& -30{y}^{4}\end{array}\right]$

$\mathbf{\text{H}}f\left(1,2\right)=\left[\begin{array}{cc}6\left(1\right)& -2\\ -2& -30\left(2{\right)}^{4}\end{array}\right]=\left[\begin{array}{cc}6& -2\\ -2& -480\end{array}\right]$

$\text{det}\left(\left[\begin{array}{cc}6& -2\\ -2& -480\end{array}\right]\right)=6\left(-480\right)-\left(-2\right)\left(-2\right)=-2884$