I have the following problem. Let $\textbf{Y} \in \mathbb{R}^{n \times q} \; n>q, \textbf{H} \in \mathbb{R}^{n \times n}$ such that $\textbf{H}$ is idempotent ($\textbf{H}^{2} = \textbf{H}$) and symmetric, positive definite whose trace is $p_{\gamma}$ where $0 \leq p_{\gamma} < q < n$. Furthermore let's assume that the product $$\textbf{Y}^\top \textbf{Y}$$ is invertible.
I'd like to know whether or not I'd be able to find the trace of the following matrix product: $$(\textbf{Y}^\top \textbf{H} \textbf{Y}) (\textbf{Y}^\top \textbf{Y})^{-1}$$ And if not, whether I'd be able to bound it in some way given what I already know.
Thanks in advance