Product of square matrices

Question

I have a square, symmetric and positive definite $n \times n$ matrix $X$. Given two $n \times n$ square matrices, $A$ and $B$, I would like to understand if there are sufficient conditions, other than $A = B$, that guarantee that the product

$Y = AXB'$

is still a square, symmetric and positive definite matrix. This would help me define a model for $Y$ in a regression model for matrices.

Answer 1

There are more general ways of producing psd's from a psd. For instance, one could consider maps

$$X \mapsto \sum_k A_k\cdot X \cdot A_k^{t}$$ for some $(A_k)_k$

There are other linear maps, the term is "positive linear map"