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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.
