Hessova matrika (oznaka
H
{\displaystyle 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
Za realno funkcijo
f
(
x
1
,
x
2
,
…
,
x
n
)
{\displaystyle f(x_{1},x_{2},\dots ,x_{n})\,}
za katero obstajajo parcialni odvodi je Hessova matrika enaka
H
(
f
)
i
j
(
x
)
=
D
i
D
j
f
(
x
)
{\displaystyle H(f)_{ij}(x)=D_{i}D_{j}f(x)\,}
kjer je
x
=
(
x
1
,
x
2
,
…
,
x
n
)
{\displaystyle x=(x_{1},x_{2},\dots ,x_{n})\,}
D
i
{\displaystyle D_{i}\,}
operator odvajanja
Hessova matrika je tako
H
(
f
)
=
[
∂
2
f
∂
x
1
2
∂
2
f
∂
x
1
∂
x
2
⋯
∂
2
f
∂
x
1
∂
x
n
∂
2
f
∂
x
2
∂
x
1
∂
2
f
∂
x
2
2
⋯
∂
2
f
∂
x
2
∂
x
n
⋮
⋮
⋱
⋮
∂
2
f
∂
x
n
∂
x
1
∂
2
f
∂
x
n
∂
x
2
⋯
∂
2
f
∂
x
n
2
]
{\displaystyle 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}\,}
Značilnosti
Jacobijeva matrika gradienta funkcije
f
{\displaystyle f\,}
je enaka Hessovi matriki, kar lahko napišemo kot
H
(
x
)
=
J
(
∇
f
)
{\displaystyle H(x)=J(\nabla f)\,}
.
V Hessovi matriki mešani odvodi funkcije
f
{\displaystyle f\,}
ležijo zunaj glavne diagonale . Ker pa zaporedje odvajanja ni pomembno, lahko zapišemo tudi
∂
∂
x
(
∂
f
∂
y
)
=
∂
∂
y
(
∂
f
∂
x
)
.
{\displaystyle {\frac {\partial }{\partial x}\left({\frac {\partial f}{\partial y}\right)={\frac {\partial }{\partial y}\left({\frac {\partial f}{\partial x}\right).}
oziroma
f
y
x
=
f
x
y
.
{\displaystyle f_{yx}=f_{xy}.\,}
.
To pomeni, da je v primerih, ko je
f
{\displaystyle f\,}
zvezna v okolici točke
D
{\displaystyle D\,}
Hessova matrika simetrična .
Če je gradient funkcije
f
{\displaystyle f\,}
v neki točki
x
{\displaystyle x\,}
enak 0, potem tej točki pravimo kritična ali stacionarna točka . Determinanta Hessove matrike se v tem primeru imenuje diskriminanta .
Omejena Hessova matrika
Omejena Hessova matrika se uporablja v nekaterih optimizacijskih problemih.
Naj bo dana funkcija
f
(
x
1
,
x
2
,
…
,
x
n
)
,
{\displaystyle f(x_{1},x_{2},\dots ,x_{n}),}
,
dodamo ji omejitveno funkcijo
g
(
x
1
,
x
2
,
…
,
x
n
)
=
c
,
{\displaystyle g(x_{1},x_{2},\dots ,x_{n})=c,}
.
V tem primeru dobimo za Hessovo matriko
H
(
f
,
g
)
=
[
0
∂
g
∂
x
1
∂
g
∂
x
2
⋯
∂
g
∂
x
n
∂
g
∂
x
1
∂
2
f
∂
x
1
2
∂
2
f
∂
x
1
∂
x
2
⋯
∂
2
f
∂
x
1
∂
x
n
∂
g
∂
x
2
∂
2
f
∂
x
2
∂
x
1
∂
2
f
∂
x
2
2
⋯
∂
2
f
∂
x
2
∂
x
n
⋮
⋮
⋮
⋱
⋮
∂
g
∂
x
n
∂
2
f
∂
x
n
∂
x
1
∂
2
f
∂
x
n
∂
x
2
⋯
∂
2
f
∂
x
n
2
]
{\displaystyle 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}
.
Glej tudi
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