黑塞矩陣

對於一元函数${\displaystyle f(x)\,}$，在給定區間內某${\displaystyle x=x_{0}\,}$點處可導，並在${\displaystyle x=x_{0}\,}$點處取得極值，其必要條件是

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

即函數${\displaystyle f(x)\,}$的極值必定在駐點處取得，或者說可導函數${\displaystyle f(x)\,}$的極值點必定是駐點；但反過來，函數的駐點不一定是極值點。檢驗駐點是否為極值點，可以採用二階導數的正負號來判斷。根據函數${\displaystyle f(x)\,}$在${\displaystyle x=x_{0}\,}$點處的泰勒展開式，考慮到上述極值必要條件，有

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

若${\displaystyle f(x)\,}$在${\displaystyle x=x_{0}\,}$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

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

即要求

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

亦即要求

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

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

設一元函数${\displaystyle f(x)\,}$在${\displaystyle x=x_{0}\,}$點處具有二階導數，且${\displaystyle f'(x_{0})=0\,}$，${\displaystyle f''(x_{0})\neq 0\,}$，則

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

而當${\displaystyle f''(x_{0})=0\,}$時，無法直接判斷，還需要逐次檢驗其更高階導數的正負號。由此有一个規律：若其開始不為零的導數階數為偶數，則駐點是極值點；若為奇數，則為拐點，而不是極值點。

對於二元函数${\displaystyle f(x_{1},x_{2})\,}$，在給定區域內某${\displaystyle x_{0}(x_{10},x_{20})\,}$點處可導，並在${\displaystyle x_{0}(x_{10},x_{20})\,}$點處取得極值，其必要條件是

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

即

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

同樣，這只是必要條件，要進一步判斷${\displaystyle x_{0}(x_{10},x_{20})\,}$是否為極值點需要找到取得極值的充分條件。根據函數${\displaystyle f(x_{1},x_{2})\,}$在${\displaystyle x_{0}(x_{10},x_{20})\,}$點處的泰勒展開式，考慮到上述極值必要條件，有

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

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

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

或

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

若${\displaystyle f(x_{1},x_{2})\,}$在${\displaystyle x_{0}(x_{10},x_{20})\,}$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

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

即要求

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

亦即要求${\displaystyle A>0\,}$，${\displaystyle AC-B^{2}>0\,}$

即

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

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

此條件反映了${\displaystyle f(x_{1},x_{2})\,}$在${\displaystyle x_{0}(x_{10},x_{20})\,}$點處的黑塞矩陣${\displaystyle G(x_{0})\,}$的各階主子式都大於零，即對於

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

要求

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

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

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

設二元函数${\displaystyle f(x_{1},x_{2})\,}$在${\displaystyle x_{0}(x_{10},x_{20})\,}$點的鄰域內連續且具有一階和二階連續偏導數，又有${\displaystyle f_{x_{1}}(x_{0})=f_{x_{2}}(x_{0})=0\,}$，同時令${\displaystyle A=f_{x_{1}x_{1}}(x_{0})\,}$，${\displaystyle B=f_{x_{1}x_{2}}(x_{0})\,}$，${\displaystyle C=f_{x_{2}x_{2}}(x_{0})\,}$，則

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

此外可以判斷，當${\displaystyle AC-B^{2}<0\,}$時，函數${\displaystyle f(x_{1},x_{2})\,}$在${\displaystyle x_{0}(x_{10},x_{20})\,}$點處沒有極值，此點稱爲鞍點。而當${\displaystyle AC-B^{2}=0\,}$時，無法直接判斷，對此，補充一個規律：當${\displaystyle AC-B^{2}=0\,}$時，如果有${\displaystyle A\equiv 0\,}$，那麼函數${\displaystyle f(x_{1},x_{2})\,}$在${\displaystyle x_{0}(x_{10},x_{20})\,}$有極值，且當${\displaystyle C>0\,}$有極小值，當${\displaystyle C<0\,}$有極大值。

由線性代數的知識可知，若矩陣${\displaystyle G(x_{0})\,}$滿足

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

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

則矩陣${\displaystyle G(x_{0})\,}$是正定矩陣，或者說矩陣${\displaystyle G(x_{0})\,}$正定。

若矩陣${\displaystyle G(x_{0})\,}$滿足

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

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

則矩陣${\displaystyle G(x_{0})\,}$是負定矩陣，或者說矩陣${\displaystyle G(x_{0})\,}$負定。[3]

於是，二元函數${\displaystyle f(x_{1},x_{2})\,}$在${\displaystyle x_{0}(x_{10},x_{20})\,}$點處取得極值的條件表述為：二元函數${\displaystyle f(x_{1},x_{2})\,}$在${\displaystyle x_{0}(x_{10},x_{20})\,}$點處的黑塞矩陣正定，則取得極小值；在${\displaystyle x_{0}(x_{10},x_{20})\,}$點處的黑塞矩陣負定，則取得極大值。

對於多元函數${\displaystyle f(x_{1},x_{2},\cdots ,x_{n})\,}$，若在${\displaystyle x_{0}(x_{1},x_{2},\cdots ,x_{n})\,}$點處取得極值，則極值存在的必要條件為

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

取得極小值的充分條件為

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

正定，即要求${\displaystyle G(x_{0})\,}$的各階主子式都大於零，即

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

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

$${\displaystyle \vdots }$$

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

取得極大值的充分條件為

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

負定。[4][5][6]