Let $0<a<b$, and let $A, B \in \mathbb{R}^{n \times n}$ be two positive definite symmetric matrices (that do not commute).
My question is: are the matrices $M, P \in \mathbb{R}^{2n \times 2n}$ defined by \begin{align*} M = \begin{bmatrix} bABA & aAB\\ aBA & bBAB \end{bmatrix},\qquad P = \begin{bmatrix} bA & aAB\\ aBA & bB \end{bmatrix} \end{align*} positive definite or not? (In the sense $x^\intercal M x >0$ for any nonzero $x \in \mathbb{R}^{2n}$.)
Why this question: I am trying to find a positive definite symmetric matrix $Q \in \mathbb{R}^{2n \times 2n}$ such that the product \begin{align*} Q \begin{bmatrix} 0 & A^{-1}\\ B^{-1} & 0 \end{bmatrix} \end{align*} is a symmetric matrix. My candidates at the moment are M and P, which are such that \begin{align*} M \begin{bmatrix} 0 & A^{-1}\\ B^{-1} & 0 \end{bmatrix} = \begin{bmatrix} aA & bAB\\ bBA & aB \end{bmatrix}, \qquad P \begin{bmatrix} 0 & A^{-1}\\ B^{-1} & 0 \end{bmatrix} = \begin{bmatrix} aA & bI_n\\ bI_n & aB \end{bmatrix} \end{align*} but I don't know if they are positive definite.
What I tried: I tried to find an invertible matrix $K$ such that the product $K^\intercal P K$ (or $K^\intercal M K$ ) is easier to study, since $P$ is positive definite if and only if $K^\intercal P K$ is. For example, for \begin{align*} K = \begin{bmatrix} A^{-1} & A^{-1}\\ B^{-1} & B^{-1} \end{bmatrix} \end{align*} we have \begin{align*} K^\intercal P K = \begin{bmatrix} bA^{-1} + 2a I_n + bB^{-1} & bA^{-1} - bB^{-1} \\ bA^{-1} - bB^{-1} & bA^{-1} - 2a I_n + bB^{-1} \end{bmatrix}. \end{align*} But I don't know how to proceed for this matrix either.