Let $b \in \mathbb{R}^n$, $c > 0$ and

$$\mathcal{M} := \{X\in \mathbb{R}^{n\times n} \mid b^{\top}Xb = c \}$$ 

Let $\mathcal{S}\_n$ be the set of real $n \times n$ symmetric positive definite (SPD) matrices and $Q \in \mathcal{S}\_n$. Let $\tilde{Q}$ be the projection of $Q$ onto $\mathcal{M}$. I want to know if $\tilde{Q}$ is still SPD. 

I suspect it is not necessarily so, and in that case I would like to have an analytical expression for $\hat{Q}$, the projection of $Q$ onto $\mathcal{M} \cap \mathcal{S}\_n$. Thanks!