Talk:Hessian matrix

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This would really be a lot easier to understand if we could see a visual representation.. something like

Hf(x,y) = [Partial 2 derivative with respect to x2], [partial 2 derivative yx]

         [Partial 2 derivative with respect to xy], [partial 2 derivative y2]

1 Is this correct?
2 Del
3 Examples
4 HUH??
5 Bordered Hessian
6 The Thereom is Wrong

[edit] Is this correct?

The second phrase of 'Second derivative test' (If the Hessian is positive definite...) should not be 'If the determinat of the Hessian is positive definite...' ?

A positive-definite matrix is a type of symmetric matrix. A determinant is just a real number, which may be positive or negative, but not positive definite. Follow the link. -GTBacchus^(talk) 23:12, 5 March 2006 (UTC)

[edit] Del

With regard to the del operator, is it that

$H=\nabla\otimes\nabla\cdot f$ ?

Or am I just confused? —Ben FrantzDale 08:13, 28 March 2006 (UTC)

I think that is close, but you need to transpose one of the dels as well as write f as a diagonal matrix:

$H=\nabla\otimes\nabla^T\cdot \mathrm{diag}(f) = \begin{bmatrix}\frac {\partial}{\partial x_1} \\ \frac {\partial}{\partial x_2} \\ \vdots \\ \frac {\partial}{\partial x_n}\end{bmatrix} \otimes \begin{bmatrix}\frac {\partial}{\partial x_1} & \frac {\partial}{\partial x_2} & \cdots & \frac {\partial}{\partial x_n}\end{bmatrix} \cdot \mathrm{diag}(f)$

$H = \begin{bmatrix} \frac{\partial^2}{\partial x_1^2} & \frac{\partial^2}{\partial x_1\partial x_2} & \cdots & \frac{\partial^2}{\partial x_1\partial x_n} \\ \frac{\partial^2}{\partial x_2\partial x_1} & \frac{\partial^2}{\partial x_2^2} & \cdots & \frac{\partial^2}{\partial x_2\partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial^2}{\partial x_n\partial x_1} & \frac{\partial^2}{\partial x_n\partial x_2} & \cdots & \frac{\partial^2}{\partial x_n^2} \end{bmatrix} \cdot \begin{bmatrix} f & 0 & \cdots & 0 \\ 0 & f & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & f\end{bmatrix} = \begin{bmatrix} \frac{\partial^2 f}{\partial x_1^2} & \frac{\partial^2 f}{\partial x_1\partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_1\partial x_n} \\ \frac{\partial^2 f}{\partial x_2\partial x_1} & \frac{\partial^2 f}{\partial x_2^2} & \cdots & \frac{\partial^2 f}{\partial x_2\partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial^2 f}{\partial x_n\partial x_1} & \frac{\partial^2 f}{\partial x_n\partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_n^2} \end{bmatrix}$

I'm pretty sure this is right--hope it helps. 16:56, 3 Apr 2006 (UTC)

[edit] Examples

It would be good to have at least one example of the use of Hessians in optimization problems, and perhaps a few words on the subject of applications of Hessians to statistical problems, e.g. maximization of parameters. --Smári McCarthy 16:01, 19 May 2006 (UTC)

[edit] HUH??

The hessian displayed is incorrect, it should be 1\2 of the second derivative matrix. Charlielewis 06:34, 11 December 2006 (UTC)

[edit] Bordered Hessian

Is not clear how a Bordered Hessian with more than one constrain should look like. If I knew I would fix it.. --Marra 16:07, 19 February 2007 (UTC)

[edit] The Thereom is Wrong

I had learned that Fxy=Fyx is Youngs Thereom, not Swartz —The preceding unsigned comment was added by Lachliggity (talk • contribs) 03:02, 16 March 2007 (UTC).

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Talk:Hessian matrix

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[edit] Is this correct?

[edit] Del

[edit] Examples

[edit] HUH??

[edit] Bordered Hessian

[edit] The Thereom is Wrong

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