# Interpolation and Restriction operators in Multigrid

I saw in several places that interpolation operator ($$P$$) and restriction operator ($$P^T$$) are usually transposes of each other (up to multiplication by a constant). As I understood it related to Galerkin condition ($$A^{2h}=P^TAP$$), where $$A$$ is the original matrix of the system $$Au=f$$ and $$A^{2h}$$ is a coarse version of $$A$$.

Why is it important to choose interpolation operator and restriction operator as transposes of each other?

Related question that I found is here

It's fundamentally because if you have that $$A^h$$ is a symmetric matrix, you want that $$A^{2h}=P^TA^hP$$ is also a symmetric matrix. You want this because you want to again use the same kind of mathematically properties for $$A^{2h}$$ as you already used for $$A^h$$ in showing that multigrid converges.