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What is the dimension of the space of $\{A\ {^t\!A}: A\in M(n\times n,\mathbb{R})\}$? I think it should be $n(n+1)/2$ if one knows already the dimension of the special orthogonal group, but I would love to derive the latter from the former.

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    Would you explain this intriguing comment, Martin? If you expanded and wrote an answer I would accept it.2012-05-04

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A cone is a subset of a vector space such that it is closed under addition and multiplication by non-negative scalars. Here we are in a finite-dimensional setting, which simplifies things a little.

Typically, a cone can be seen as the "positive part" of your space. Canonical examples are $[0,\infty)\subset\mathbb{R}$, $[0,\infty)\subset\mathbb{C}$, and the positive-semidefinite matrices (either real or complex); this last one is your example.

The main property of the cone is that it is convex (i.e. $x,y$ in the cone, $t\in[0,1]$ imply that $tx+(1-t)y$ is in the cone), and so it can be seen as generated (as a convex set) by its extreme points (that is, every point is a convex combination of extreme points). In your case, the extreme points are the scalar multiplies of the projections, i.e. $\lambda A$ with $\lambda\geq0$, $A=A^t=A^2$.

All this said, I have no idea if this helps with your problem!

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    Many definitions of cone that I have seen just require $\lambda x$ be in the code when $x$ is in the cone and \lambda>0. If $0$ is in the cone, it is sometimes called a pointed cone. That is, convexity is not necessarily part of the definition. Not that it matters here.2012-05-04