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The set of positive definite matrices in $\mathbb R^{n\times n}$ is geometrically a positive cone. This statement appears in almost every article on real positive definite matrices I read but without a proof. Where can I find a general proof, please? In addition, if I assume this statement is true, does it imply that the set of positive definite matrices is a manifold or a sub-manifold of $\mathbb R^n$ for some $n$?

LaTeXFan
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    What part of the definition are you having trouble verifying? – Qiaochu Yuan Apr 30 '14 at 23:11
  • I know what positive definite matrices are. And I know what a cone is, at least in $\mathbb R^3$? But I don't know why the two are equivalent which seems true from the statement I quoted. – LaTeXFan Apr 30 '14 at 23:15
  • What definition of a cone are you working with? – Qiaochu Yuan Apr 30 '14 at 23:17
  • In fact, I want to show that the set of all inner products in $\mathbb R^3$ is some manifold and determine its dimension. I learnt that these inner products are equivalent to the set of positive definite matrices. After that I read the statement I quoted. So I do not know the definition of cone used in this statement. – LaTeXFan Apr 30 '14 at 23:23
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    I believe the definition in question may be that of a convex cone - and I think it's pretty intuitive that the set of positive definite matrices satisfies it (positive definite matrices correspond to positive quadratic forms, and positive linear combination of those is obviously positive). – Marcin Łoś Apr 30 '14 at 23:48

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It's immediate from the definitions. An $n\times n$ real matrix $A$ is positive definite if (1) it is symmetric ($A^t=A$) and (2) $v^tAv>0$ for all non-zero column vectors $v\in\mathbb R^n$. The 1st condition is linear, so defines a linear subspace of $\mathbb R^{n^2},$ of dimension $n(n+1)/2$ (easy exercise). The 2nd condition defines an open subset of $\mathbb R^{n(n+1)/2}$ which is a convex cone; ie a subset closed under linear combinations with positive coefficients. This is very easy to verify. To show that its open you use the fact that $(v,A)\mapsto v^tAv$ is a continuous function. Thus your set is an $n(n+1)/2$ dimensional manifold, a convex open subset of $\mathbb R^{n(n+1)/2}$.

Gil Bor
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