Author Topic: Critical Points Analysis - Hessian Matrix  (Read 2183 times)

Offline Alberto Dominguez

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Critical Points Analysis - Hessian Matrix
« on: July 21, 2011, 01:56:59 PM »
f:\mathbb{R}^{n}\longrightarrow \mathbb{R}

Critical Points:

\frac{\partial f}{\partial x_{i}}=0

Second order partial derivative test:

Hessian: H{}_{f}(x)_{i,j}=\frac{\partial^{2}f}{\partial x_{i}\partial x_{j}}

H(f)=\left[\begin{array}{cccc}<br />\frac{\partial^{2}f}{\partial x_{1}^2} & \frac{\partial^{2}f}{\partial x_{1}\partial x_{2}} & \ldots & \frac{\partial^{2}f}{\partial x_{1}\partial x_{n}}\\<br />\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}}\\<br />\vdots & \vdots & \ddots & \vdots\\<br />\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{array}\right]

Classification by Eigenvalues criteria:

1.- H(f) Positive Definite - \lambda_{i}>0 - Minimum
2.- H(f) Negative Definite - \lambda_{i}<0 - Maximum
3.- H(f) Indefinite - \lambda_{i_{1}}>0,\:\lambda_{i_{2}}<0 - Saddle point
4.- H(f) Positive Semi-Definite - \lambda_{i}\geq0 - No information
5.- H(f) Negative Semi-Definite - \lambda_{i}\leq0 - No information
« Last Edit: July 21, 2011, 09:41:02 PM by Alberto Dominguez »
Kind regards,

Alberto Domínguez
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