# Find the dimension of the cone of positive definite matrices

Let $$V$$, $$W$$ be the $$n$$, $$m$$ dimensional real vector spaces with scalar product.

What is the dimension of the cones $$P(V)$$, $$P(Votimes W)$$ of positive definite complex operators on $$V$$, $$Votimes W$$ with traces equal to 1?