If I am not wrong, the set of definite positive matrices with real coefficients is a convex cone without the vertex, which is the null matrix. What is the number of degrees of freedom for this set of matrices?
Please apologize me for not being formal. I will try to explain better my question.
In statistics, a positive definite matrix has a special meaning, since it is a variance-covariance matrix. In statistical inference, we want to have a number of samples that it is at least equal to the number of degrees of freedom of the parameter space. Suppose to have a set of observation from a multivariate normal model $N(\mathbf{0}, \mathbf{V})$, and that you want to do inference on $\mathbf{V}$. You need a number of samples at least equal to the number of degrees of freedom for the matrix $\mathbf{V}$.
So, my question, how many degrees of freedom has the matrix $\mathbf{V}$? How to formalize this concept? Is "number degrees of freedom" a synonym for "dimension of a manifold" or something similar? Thank you for the clarification.
One usually thinks of $n\times n$ dimensional positive definite matrices as being an open subset of the $n\times n$ symmetric matrices. This latter space is a $n+\binom{n}2$-dimensional linear manifold. The p.d. matrices form a manifold of this dimension.
"Degrees of freedom", if it means anything in this context, means dimension or more often (in the context of hypothesis testing, as with chi-squared tests, $F$-tests, and various applications of Wilks's Theorem) codimension of a submanifold of parameters (the "the restricted model hypothesis") imbedded in another ("the general model hypothesis").