For which values the matrix $ B = \Big(\begin{matrix} A & -A\\ -A & \alpha A \end{matrix}\Big)$ results positive definite (strict)?

Question

I'm trying to solve the next problem where given $A \in \mathbb{R}^{n \times n}$ symmetric positive definite I have to find the values for $\alpha \in \mathbb{R}$ such that this matrix:

$ B = \Big(\begin{matrix} A & -A\\ -A & \alpha A \end{matrix}\Big)$

Is positive definite. Also I notice that $B$ is symmetric, so I tried to find it's Cholesky's decomposition by block product, but I wasn't able to do so.

Answer 1

$$ \left( \begin{array}{rr} I & I \\ I & 0 \\ \end{array} \right) \left( \begin{array}{rr} A & -A \\ -A & \alpha A \\ \end{array} \right) \left( \begin{array}{rr} I & I \\ I & 0 \\ \end{array} \right) = \left( \begin{array}{cc} (\alpha-1)A & 0 \\ 0& A \\ \end{array} \right) $$

Answer 2

Another approach: this matrix can be written as the Kronecker product $$ B = \pmatrix{1&-1\\-1 & \alpha} \otimes A. $$ It follows that $B$ will be positive definite if and only if $\pmatrix{1&-1\\-1&\alpha}$ is positive definite.