# Significance of Hessian Matrix [duplicate]

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I want to know the geometrical significance of Hessian Matrix. Please could anyone have any idea about it?

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## marked as duplicate by Giuseppe Negro, azimut, Rick Decker, TZakrevskiy, Davide GiraudoSep 16 '13 at 13:14

It is unclear to me what the question is. Are you aware of the relationship between the Hessian and convexity? Can you please be more specific? – Jonas Meyer Jul 12 '12 at 6:40
A sci-fi film set in the 18th century with German mercenaries. – copper.hat Jul 12 '12 at 6:41
@copper, no, it's what I'd use to trap Hessian flies... nasty buggers. – J. M. Jul 12 '12 at 6:48
'Sisal'ing observation! Hessian flies were supposedly imported by the Hessians (it says so on wiki, so it must be true...). – copper.hat Jul 12 '12 at 6:51
I must add, copper.hat, that comment is probably the best I've seen on this site. – Jeff Jul 12 '12 at 8:21

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").