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$X_{n \times p}$ 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 $(X^TX)^{-1}$?

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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

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