Let
$$ M = \begin{bmatrix} C \\ CA \\ CA^2 \\ \vdots \\ CA^k\end{bmatrix}\\ Y = \begin{bmatrix} y_0 \\ \vdots \\ y_k\end{bmatrix} $$
Is there a convenient form to compute $MM^\dagger Y$ with $M^\dagger$ the pseudoinverse of $M$?
If it helps, I can also assume that $A$ is diagonal or $C$ is fat with an identity (although not both): $$ C = \begin{bmatrix} I & 0 \end{bmatrix} $$
Edit: I think computing $MM^\dagger$ amounts to computing $U\Sigma V^*V\Sigma^{-1}U^* = UU^*$ and so if there was a "quick" way to compute the left eigenvectors of $M$ then computing $UU^*$ would be likewise quick.