Is there an interpretation of $X^{\dagger}Y$ in terms of a projection or a least-squares formulation?
Note that $\dagger$ denotes the pseudo-inverse, $X$ is a square real matrix and $Y$ is a rectangular real matrix. Am trying to have a better interpretation of the pseudo-inverse in this situation.
What does the pseudoinverse of $A$ do? It takes a vector $b$ as input, and returns as output the vector $x$ of least 2-norm such that $Ax = \hat{b}$, where $\hat{b}$ is the projection of $b$ onto the column space of $A$.
Strang's book Linear Algebra and Its Applications has a good presentation of this topic.