PlanetMath: Hessian matrix

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Hessian matrix

(Definition)

Let $x \in \mathbb{R}^n$ and let $f\colon\mathbb{R}^n\to\mathbb{R}$ be a real-valued function having 2nd-order partial derivatives in an open set containing . The Hessian matrix of is the matrix of second partial derivatives evaluated at :

$\displaystyle \mathbf{H}(x):= \begin{bmatrix}\displaystyle{\frac{\partial^2 f}{... ...}} & \ldots & \displaystyle{\frac{\partial^2 f}{\partial x_n^2}} \end{bmatrix}.$

(1)

If is in , $\mathbf{H}(x)$ is symmetric because of the equality of mixed partials. Note that $\mathbf{H}(x)=\mathbf{J}(\nabla f)$ , the Jacobian of the gradient of .

Given a vector $\boldsymbol{v}\in\mathbb{R}^n$ , the Hessian of at $\boldsymbol{v}$ is:

$\displaystyle \mathbf{H}(x)(\boldsymbol{v}):=\frac{1}{2}\boldsymbol{v}\mathbf{H}(x)\boldsymbol{v}^{\operatorname{T}}.$

(2)

Here we view $\boldsymbol{v}$ as a

matrix so that $\boldsymbol{v}^{\operatorname{T}}$ is the transpose of $\boldsymbol{v}$ .

Remark. The Hessian of at $\boldsymbol{v}$ is a quadratic form, since $\mathbf{H}(x)(r\boldsymbol{v})=r^2\mathbf{H}(x)(\boldsymbol{v})$ for any $r\in\mathbb{R}$ .

If is further assumed to be in , and is a critical point of such that $\mathbf{H}(x)$ is positive definite, then is a strict local minimum of .

This is not difficult to show. Since $\mathbf{H}(x)$ is positive definite, the Rayleigh-Ritz theorem shows that there is a such that for all $h \in \mathbb{R}^n$ , $h^T\mathbf{H}(x)h \ge 2c \Vert h\Vert^2$ . Thus by Taylor's theorem (weaker form)

$\displaystyle f(x + h ) = f(x) + \frac{1}{2} h^T\mathbf{H}(x)h + o(\Vert h \Vert^2) \ge c \Vert h \Vert^2 + o(\Vert h\Vert^2).$

For small $\Vert h \Vert$ the first term on the right dominates the second, so that both sides are positive for small $\Vert h\Vert$ .

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"Hessian matrix" is owned by cvalente. [ full author list (6) | owner history (1) ]

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Also defines:

Hessian

Keywords:

second derivative

Attachments:: relations between Hessian matrix and local extrema (Result) by bshanks

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Cross-references: positive, sides, Rayleigh-Ritz theorem, strict local minimum, critical point, quadratic form, transpose, vector, gradient, Jacobian, equality of mixed partials, matrix, open set, partial derivatives, function
There are 11 references to this entry.

This is version 28 of Hessian matrix, born on 2002-08-28, modified 2007-04-19.
Object id is 3370, canonical name is HessianMatrix.
Accessed 41831 times total.

Classification:

AMS MSC:

26B12 (Real functions :: Functions of several variables :: Calculus of vector functions)

Pending Errata and Addenda

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