Suppose a symmetric positive definite $A \in \mathbb{R}^{d \times d}$, satisfies $\gamma I \preceq A \preceq \Gamma I$, for some $0<\gamma \leq \Gamma$.
Given a positive integer $k<d$, let $v\in \mathbb{R}^d$ be the diagonal terms of $A$, that is $v = \text{diag(A)}$, and $v_k$ be the $k$-th largest value of $v$.
Define a sampling matrix $S$ as $S = I + H_k$, where $H_k$ is defined as \begin{equation} H_k[i,j] = \begin{cases} 1, & \left( (v[i] \geq v_k) \wedge (v[j] \geq v_k) \wedge (i \neq j) \right) \\ 0, & \text{others} \end{cases} \end{equation}
The corresponding approximate matrix $B$ is defined as $B = S \cdot A$, where $\cdot$ is element-wise product.
Is $B$ still positive definite matrix? If so, $\gamma I \preceq B \preceq \Gamma I$ or other inequality holds?