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 odvajanjaHessova 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}}} .
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