I want to know the geometrical significance of Hessian Matrix. Please could anyone have any idea about it?
Significance of Hessian Matrix
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multivariable-calculus
derivatives
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1'Sisal'ing observation! Hessian flies were supposedly imported by the Hessians (it says so on wiki, so it must be true...). – 2012-07-12
1 Answers
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Basically, it's a symmetric matrix (Young's theorem) used to describe curvature for functions of a vector variable. For a real valued function of a vector variable, $f:\mathbb{R^n}\rightarrow\mathbb{R}$ it's $n\times n$. The results of $uHu$ are of interest for optimization problems because the Hessian serves to describe local behavior of the function at those points (much like the second derivative test works for $\mathbb{R}\rightarrow\mathbb{R}$). Traits of the eigenvalues of the Hessian also do this (i.e. "positive definite").
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0as said in your answer, Hessian serves to describe local behavior of function, Hessian is just the quadratic term in Taylor expansion, so how could it alone describe the local behavior without taking 1st order derivative into account? – 2013-09-06