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单词 HessianMatrix
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Hessian matrix


Let xn and let f:n be a real-valued function having 2nd-order partial derivativesMathworldPlanetmath in an open set U containing x. The Hessian matrix of f is the matrix of second partial derivatives evaluated at x:

𝐇(x):=[2fx122fx1x22fx1xn2fx2x12fx222fx2xn2fxnx12fxnx22fxn2].(1)

If f is in C2(U), 𝐇(x) is symmetricPlanetmathPlanetmath (http://planetmath.org/SymmetricMatrix) because of the equality of mixed partials. Note that 𝐇(x)=𝐉(f), the JacobianDlmfMathworldPlanetmathPlanetmath of the gradientMathworldPlanetmath of f.

Given a vector 𝒗n, the Hessian of f at 𝒗 is:

𝐇(x)(𝒗):=12𝒗𝐇(x)𝒗T.(2)

Here we view 𝒗 as a 1 by n matrix so that 𝒗T is the transposeMathworldPlanetmath of 𝒗.

Remark. The Hessian of f at 𝒗 is a quadratic formMathworldPlanetmath, since 𝐇(x)(r𝒗)=r2𝐇(x)(𝒗) for any r.

If f is further assumed to be in C2(U), and x is a critical pointDlmfMathworld of f such that 𝐇(x) is positive definitePlanetmathPlanetmath (http://planetmath.org/PositiveDefinite), thenx is a strict local minimum of f.

This is not difficult to show. Since 𝐇(x) is positive definite (http://planetmath.org/PositiveDefinite), the Rayleigh-Ritz theorem shows that there is a c>0 such that for all hn,hT𝐇(x)h2ch2. Thus byTaylor’s theorem (http://planetmath.org/TaylorPolynomialsInBanachSpaces) ( form)

f(x+h)=f(x)+12hT𝐇(x)h+o(h2)ch2+o(h2).

For small h the first on the the second, so that both sides are positive for small h.

TitleHessian matrix
Canonical nameHessianMatrix
Date of creation2013-03-22 12:59:41
Last modified on2013-03-22 12:59:41
Ownercvalente (11260)
Last modified bycvalente (11260)
Numerical id31
Authorcvalente (11260)
Entry typeDefinition
Classificationmsc 26B12
Related topicGradient
Related topicPartialDerivative
Related topicSymmetricMatrix
Related topicComplexHessianMatrix
Related topicHessianForm
Related topicDirectionalDerivative
DefinesHessian
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更新时间:2025/5/3 13:31:45