Rank of quadratic matrix form

Question

Let $p > r$. Suppose I have a $p \times r$ matrix $V$ of rank $r$ and a $p \times p$ positive semi-definite matrix $X$ of rank $q$ where $q < p$. What does this imply about the rank of $$ V^T X V?$$

Is this matrix positive definite if $r < q$?

Answer 1

Since $X$ has a rank $q$, we can write $X = L^TL$, where $L \in \mathbb{R}^{q \times p}$. Hence, we have $$V^TXV = (LV)^T (LV) = Y^TY$$ where $Y \in \mathbb{R}^{q \times r}$. Hence, the rank of the matrix is bounded above by $\min(q,r)$. However, the matrix need not be full-rank, i.e., it needn't be positive definite. For instance, consider the following example. $$X = \begin{bmatrix} 1 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0\end{bmatrix} \text{ and }V = \begin{bmatrix} 0 & 0\\ 1 & 0\\ 0 & 2\end{bmatrix}$$ We then have $$V^TXV = \begin{bmatrix}0 & 0\\ 0 & 0\end{bmatrix}$$ which is clearly not positive definite.