# Moore-Penrose Pseudoinverse of a matrix product

$$X_{n \times p}$$ is a real, thin ($$n>p$$) rectangular matrix of rank $$p$$, so $$X^T X$$ is full rank. The Moore-Penrose pseudoinverse of $$X$$ is given by $$X^+=(X^TX)^{-1}X^T$$.

Let's now define $$W=XA$$

($$X$$ is linearly transformed by $$A_{p \times k}$$ to produce a $$W_{n \times k}$$). $$W$$ is of rank $$k$$ (so $$W^TW$$ is full rank).

I need to compute the Moore-Penrose Pseudoinverse of $$W$$ for many different $$A$$ matrices efficiently. Is there an identity or a decomposition that allows expressing $$W^+$$ as some function of $$X^+$$ or $$(X^TX)^{-1}$$?

due to distributive order reversal rule for transposing multiple and associativity of multiplication , $$W^+=(A^T (X^TX) A)^{-1} A^T X^T$$ not too much opportunity for pre-calculation it seems