# 黑塞矩陣

（重定向自海森矩阵

## 定義

${\displaystyle \mathbf {H} ={\begin{bmatrix}{\frac {\partial ^{2}f}{\partial x_{1}^{2}}}&{\frac {\partial ^{2}f}{\partial x_{1}\,\partial x_{2}}}&\cdots &{\frac {\partial ^{2}f}{\partial x_{1}\,\partial x_{n}}}\\\\{\frac {\partial ^{2}f}{\partial x_{2}\,\partial x_{1}}}&{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}&\cdots &{\frac {\partial ^{2}f}{\partial x_{2}\,\partial x_{n}}}\\\\\vdots &\vdots &\ddots &\vdots \\\\{\frac {\partial ^{2}f}{\partial x_{n}\,\partial x_{1}}}&{\frac {\partial ^{2}f}{\partial x_{n}\,\partial x_{2}}}&\cdots &{\frac {\partial ^{2}f}{\partial x_{n}^{2}}}\end{bmatrix}}\,}$

${\displaystyle \mathbf {H} _{ij}={\frac {\partial ^{2}f}{\partial x_{i}\partial x_{j}}}}$

## 性質

${\displaystyle f(x)=f(x_{0})+f'(x_{0})\Delta x+{\frac {f''(x_{0})}{2!}}\Delta x^{2}+\cdots \,}$

${\displaystyle f(x_{1},x_{2})=f(x_{10},x_{20})+f_{x_{1}}(x_{10},x_{20})\Delta x_{1}+f_{x_{2}}(x_{10},x_{20})\Delta x_{2}+{\frac {1}{2}}[f_{x_{1}x_{1}}(x_{10},x_{20})\Delta x_{1}^{2}+2f_{x_{1}x_{2}}(x_{10},x_{20})\Delta x_{1}\Delta x_{2}+f_{x_{2}x_{2}}(x_{10},x_{20})\Delta x_{2}^{2}]+\cdots \,}$

${\displaystyle f(x)=f(x_{0})+\nabla f(x_{0})^{\mathrm {T} }\Delta x+{\frac {1}{2}}\Delta x^{\mathrm {T} }G(x_{0})\Delta x+\cdots }$

${\displaystyle G(x_{0})={\begin{bmatrix}{\frac {\partial ^{2}f}{\partial x_{1}^{2}}}&{\frac {\partial ^{2}f}{\partial x_{1}\,\partial x_{2}}}\\\\{\frac {\partial ^{2}f}{\partial x_{2}\,\partial x_{1}}}&{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}\end{bmatrix}}_{x_{0}}\,}$

${\displaystyle {\frac {\partial ^{2}f}{\partial x_{1}\partial x_{2}}}={\frac {\partial ^{2}f}{\partial x_{2}\partial x_{1}}}}$

${\displaystyle f(x)=f(x_{0})+\nabla f(x_{0})^{\mathrm {T} }\Delta x+{\frac {1}{2}}\Delta x^{\mathrm {T} }G(x_{0})\Delta x+\cdots \,}$

${\displaystyle \nabla f(x_{0})={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}&{\frac {\partial f}{\partial x_{2}}}&\cdots &{\frac {\partial f}{\partial x_{n}}}\end{bmatrix}}_{x_{0}}^{T}\,}$

${\displaystyle G(x_{0})={\begin{bmatrix}{\frac {\partial ^{2}f}{\partial x_{1}^{2}}}&{\frac {\partial ^{2}f}{\partial x_{1}\,\partial x_{2}}}&\cdots &{\frac {\partial ^{2}f}{\partial x_{1}\,\partial x_{n}}}\\\\{\frac {\partial ^{2}f}{\partial x_{2}\,\partial x_{1}}}&{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}&\cdots &{\frac {\partial ^{2}f}{\partial x_{2}\,\partial x_{n}}}\\\\\vdots &\vdots &\ddots &\vdots \\\\{\frac {\partial ^{2}f}{\partial x_{n}\,\partial x_{1}}}&{\frac {\partial ^{2}f}{\partial x_{n}\,\partial x_{2}}}&\cdots &{\frac {\partial ^{2}f}{\partial x_{n}^{2}}}\end{bmatrix}}_{x_{0}}\,}$

${\displaystyle \mathrm {H} (f)=\mathrm {J} (\nabla f)\,}$

## 應用

### 函數的極值條件

${\displaystyle f'(x_{0})=0\,}$

${\displaystyle f(x)=f(x_{0})+{\frac {f''(x_{0})}{2!}}\Delta x^{2}+\cdots \,}$

${\displaystyle f(x)\,}$ ${\displaystyle x=x_{0}\,}$ 點處取得極小值，則要求在${\displaystyle x=x_{0}\,}$ 某一鄰域內一切點${\displaystyle x\,}$ 都必須滿足

${\displaystyle f(x)-f(x_{0})>0\,}$

${\displaystyle {\frac {f''(x_{0})}{2!}}\Delta x^{2}>0\,}$

${\displaystyle f''(x_{0})>0\,}$

${\displaystyle f(x)\,}$ ${\displaystyle x=x_{0}\,}$ 點處取得極大值的討論與之類似。於是有極值充分條件

1. ${\displaystyle f''(x_{0})>0\,}$ 時，函數${\displaystyle f(x)\,}$ ${\displaystyle x=x_{0}\,}$ 處取得極小值；
2. ${\displaystyle f''(x_{0})<0\,}$ 時，函數${\displaystyle f(x)\,}$ ${\displaystyle x=x_{0}\,}$ 處取得極大值。

${\displaystyle f_{x_{1}}(x_{0})=f_{x_{2}}(x_{0})=0\,}$

${\displaystyle \nabla f(x_{0})=0\,}$

${\displaystyle f(x_{1},x_{2})=f(x_{10},x_{20})+{\frac {1}{2}}[f_{x_{1}x_{1}}(x_{0})\Delta x_{1}^{2}+2f_{x_{1}x_{2}}(x_{0})\Delta x_{1}\Delta x_{2}+f_{x_{2}x_{2}}(x_{0})\Delta x_{2}^{2}]+\cdots \,}$

${\displaystyle A=f_{x_{1}x_{1}}(x_{0})\,}$ ${\displaystyle B=f_{x_{1}x_{2}}(x_{0})\,}$ ${\displaystyle C=f_{x_{2}x_{2}}(x_{0})\,}$ ，則

${\displaystyle f(x_{1},x_{2})=f(x_{10},x_{20})+{\frac {1}{2}}[A\Delta x_{1}^{2}+2B\Delta x_{1}\Delta x_{2}+C\Delta x_{2}^{2}]+\cdots \,}$

${\displaystyle f(x_{1},x_{2})=f(x_{10},x_{20})+{\frac {1}{2A}}[(A\Delta x_{1}+B\Delta x_{2})^{2}+(AC-B^{2})\Delta x_{2}^{2}]+\cdots \,}$

${\displaystyle f(x_{1},x_{2})\,}$ ${\displaystyle x_{0}(x_{10},x_{20})\,}$ 點處取得極小值，則要求在${\displaystyle x_{0}(x_{10},x_{20})\,}$ 某一鄰域內一切點${\displaystyle x\,}$ 都必須滿足

${\displaystyle f(x_{1},x_{2})-f(x_{10},x_{20})>0\,}$

${\displaystyle {\frac {1}{2A}}[(A\Delta x_{1}+B\Delta x_{2})^{2}+(AC-B^{2})\Delta x_{2}^{2}]>0\,}$

${\displaystyle \left.{\frac {\partial ^{2}f}{\partial x_{1}^{2}}}\right|_{x_{0}}>0\,}$

${\displaystyle {\begin{bmatrix}{\frac {\partial ^{2}f}{\partial x_{1}^{2}}}{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}-({\frac {\partial ^{2}f}{\partial x_{1}\partial x_{2}}})^{2}\end{bmatrix}}_{x_{0}}>0\,}$

${\displaystyle G(x_{0})={\begin{bmatrix}{\frac {\partial ^{2}f}{\partial x_{1}^{2}}}&{\frac {\partial ^{2}f}{\partial x_{1}\,\partial x_{2}}}\\\\{\frac {\partial ^{2}f}{\partial x_{2}\,\partial x_{1}}}&{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}\end{bmatrix}}_{x_{0}}\,}$

${\displaystyle \left.{\frac {\partial ^{2}f}{\partial x_{1}^{2}}}\right|_{x_{0}}>0\,}$

${\displaystyle |G(x_{0})|={\begin{vmatrix}{\frac {\partial ^{2}f}{\partial x_{1}^{2}}}&{\frac {\partial ^{2}f}{\partial x_{1}\,\partial x_{2}}}\\\\{\frac {\partial ^{2}f}{\partial x_{2}\,\partial x_{1}}}&{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}\end{vmatrix}}_{x_{0}}>0\,}$

${\displaystyle f((x_{1},x_{2})\,}$ ${\displaystyle x_{0}(x_{10},x_{20})\,}$ 點處取得極大值的討論與之類似。於是有極值充分條件：

1. ${\displaystyle A>0\,}$ ${\displaystyle AC-B^{2}>0\,}$ 時，函數${\displaystyle f(x_{1},x_{2})\,}$ ${\displaystyle x_{0}(x_{10},x_{20})\,}$ 處取得極小值；
2. ${\displaystyle A<0\,}$ ${\displaystyle AC-B^{2}>0\,}$ 時，函數${\displaystyle f(x_{1},x_{2})\,}$ ${\displaystyle x_{0}(x_{10},x_{20})\,}$ 處取得極大值。

${\displaystyle \left.{\frac {\partial ^{2}f}{\partial x_{1}^{2}}}\right|_{x_{0}}>0\,}$

${\displaystyle {\begin{vmatrix}{\frac {\partial ^{2}f}{\partial x_{1}^{2}}}&{\frac {\partial ^{2}f}{\partial x_{1}\,\partial x_{2}}}\\\\{\frac {\partial ^{2}f}{\partial x_{2}\,\partial x_{1}}}&{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}\end{vmatrix}}_{x_{0}}>0\,}$

${\displaystyle \left.{\frac {\partial ^{2}f}{\partial x_{1}^{2}}}\right|_{x_{0}}<0\,}$

${\displaystyle {\begin{vmatrix}{\frac {\partial ^{2}f}{\partial x_{1}^{2}}}&{\frac {\partial ^{2}f}{\partial x_{1}\,\partial x_{2}}}\\\\{\frac {\partial ^{2}f}{\partial x_{2}\,\partial x_{1}}}&{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}\end{vmatrix}}_{x_{0}}>0\,}$

${\displaystyle \nabla f(x_{0})={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}&{\frac {\partial f}{\partial x_{2}}}&\cdots &{\frac {\partial f}{\partial x_{n}}}\end{bmatrix}}_{x_{0}}^{T}=0\,}$

${\displaystyle G(x_{0})={\begin{bmatrix}{\frac {\partial ^{2}f}{\partial x_{1}^{2}}}&{\frac {\partial ^{2}f}{\partial x_{1}\,\partial x_{2}}}&\cdots &{\frac {\partial ^{2}f}{\partial x_{1}\,\partial x_{n}}}\\\\{\frac {\partial ^{2}f}{\partial x_{2}\,\partial x_{1}}}&{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}&\cdots &{\frac {\partial ^{2}f}{\partial x_{2}\,\partial x_{n}}}\\\\\vdots &\vdots &\ddots &\vdots \\\\{\frac {\partial ^{2}f}{\partial x_{n}\,\partial x_{1}}}&{\frac {\partial ^{2}f}{\partial x_{n}\,\partial x_{2}}}&\cdots &{\frac {\partial ^{2}f}{\partial x_{n}^{2}}}\end{bmatrix}}_{x_{0}}\,}$

${\displaystyle \left.{\frac {\partial ^{2}f}{\partial x_{1}^{2}}}\right|_{x_{0}}>0\,}$

${\displaystyle {\begin{vmatrix}{\frac {\partial ^{2}f}{\partial x_{1}^{2}}}&{\frac {\partial ^{2}f}{\partial x_{1}\,\partial x_{2}}}\\\\{\frac {\partial ^{2}f}{\partial x_{2}\,\partial x_{1}}}&{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}\end{vmatrix}}_{x_{0}}>0\,}$

${\displaystyle \vdots }$

${\displaystyle |G(x_{0})|>0\,}$

${\displaystyle G(x_{0})={\begin{bmatrix}{\frac {\partial ^{2}f}{\partial x_{1}^{2}}}&{\frac {\partial ^{2}f}{\partial x_{1}\,\partial x_{2}}}&\cdots &{\frac {\partial ^{2}f}{\partial x_{1}\,\partial x_{n}}}\\\\{\frac {\partial ^{2}f}{\partial x_{2}\,\partial x_{1}}}&{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}&\cdots &{\frac {\partial ^{2}f}{\partial x_{2}\,\partial x_{n}}}\\\\\vdots &\vdots &\ddots &\vdots \\\\{\frac {\partial ^{2}f}{\partial x_{n}\,\partial x_{1}}}&{\frac {\partial ^{2}f}{\partial x_{n}\,\partial x_{2}}}&\cdots &{\frac {\partial ^{2}f}{\partial x_{n}^{2}}}\end{bmatrix}}_{x_{0}}\,}$

## 參考文獻

1. ^ Binmore, Ken; Davies, Joan. Calculus Concepts and Methods. Cambridge University Press. 2007: 190. ISBN 9780521775410. OCLC 717598615.
2. ^ 白清顺; 孙靖明; 梁迎春 (编). 机械优化设计（第6版）. 北京: 机械工业出版社. 2017.6（2018.11重印）: 35~36页. ISBN 978-7-111-56643-4.
3. ^ 刘二根; 谢霖铨 (编). 线性代数. 江西高校出版社. 2015.7: 164~166页. ISBN 978-7-5493-3588-6.
4. ^ 白清顺; 孙靖明; 梁迎春 (编). 机械优化设计（第6版）. 北京: 机械工业出版社. 2017.6（2018.11重印）: 37~39页. ISBN 978-7-111-56643-4.
5. ^ 同济大学数学系 (编). 高等数学（第七版）上册. 高等教育出版社. 2014.7: 155页. ISBN 978-7-04-039663-8.
6. ^ 同济大学数学系 (编). 高等数学（第七版）下册. 高等教育出版社. 2014.7: 113页. ISBN 978-7-04-039662-1.