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If $X$ is an $n \times p$ matrix of rank $r$ and $C = AX$ for some $q \times n$ matrix $A$ with rank$(A) = q$, how do I show that rank $(X(I-C^{-}C))=$ rank$(X)-$ rank$(C)$? I can show that rank $(X(I-C^{-}C))\geq $ rank$(X)-$ rank$(C)$, but how do I get the reverse inequality?

Note : $C^{-}$ is a generalized inverse of $C$.

Any help would be appreciated.

1 Answers 1