Hessova matrika

Hessova matrika (oznaka $H\,$ ) (tudi hesian) je kvadratna matrika, ki jo sestavljajo drugi parcialni odvodi neke funkcije.

Imenuje se po nemškem matematiku Ludwigu Ottu Hesseju (1811 – 1874), ki jo je raziskoval v 19. stoletju. Pozneje so jo poimenovali po njem.

Definicija

f(x_{1},x_{2},\dots ,x_{n})\,

za katero obstajajo parcialni odvodi je Hessova matrika enaka

H(f)_{ij}(x)=D_{i}D_{j}f(x)\,

kjer je

Hessova matrika je tako

H(f)={\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}}\,

Jacobijeva matrika gradienta funkcije $f\,$ je enaka Hessovi matriki, kar lahko napišemo kot $H(x)=J(\nabla f)\,$ .

V Hessovi matriki mešani odvodi funkcije $f\,$ ležijo zunaj glavne diagonale. Ker pa zaporedje odvajanja ni pomembno, lahko zapišemo tudi

{\frac {\partial }{\partial x}}\left({\frac {\partial f}{\partial y}}\right)={\frac {\partial }{\partial y}}\left({\frac {\partial f}{\partial x}}\right).

oziroma

f_{yx}=f_{xy}.\,

.

To pomeni, da je v primerih, ko je $f\,$ zvezna v okolici točke $D\,$ Hessova matrika simetrična.

Če je gradient funkcije $f\,$ v neki točki $x\,$ enak 0, potem tej točki pravimo kritična ali stacionarna točka. Determinanta Hessove matrike se v tem primeru imenuje diskriminanta.

Omejena Hessova matrika se uporablja v nekaterih optimizacijskih problemih. Naj bo dana funkcija

f(x_{1},x_{2},\dots ,x_{n}),

,

dodamo ji omejitveno funkcijo

g(x_{1},x_{2},\dots ,x_{n})=c,

.

V tem primeru dobimo za Hessovo matriko

H(f,g)={\begin{bmatrix}0&{\frac {\partial g}{\partial x_{1}}}&{\frac {\partial g}{\partial x_{2}}}&\cdots &{\frac {\partial g}{\partial x_{n}}}\\\\{\frac {\partial g}{\partial x_{1}}}&{\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 g}{\partial x_{2}}}&{\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 &\vdots &\ddots &\vdots \\\\{\frac {\partial g}{\partial x_{n}}}&{\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}}

.