When $\mathbb{E}[X|AX+AY]=\mathbb{E}[X|AX]$ holds?

Question

When $\mathbb{E}[X|AX+AY]=\mathbb{E}[X|AX]$ holds, $A$ is a known matrix, $X$ and $Y$ are random matrixes

Does it hold without conditions?

I can understand when $X=Y$, but I cannot imagine other cases.

Thanks!

Answer 1

$\mathbb E[X \mid A X + A Y]$ is a function of $A X + A Y = A (X+Y)$, and therefore a function of $X+Y$, while $\mathbb E[X \mid A X]$ is a function of $X$. If $A$ is invertible, $X = A^{-1}(AX)$ so by linearity $$\mathbb E[X \mid AX] = A^{-1} \mathbb E[AX \mid AX] = A^{-1} (AX) = X$$ There is no reason for these to be equal, and in nearly all examples they will be different.