I'm told this expression $(^)(^)^{-1}$ has a name and I suppose, has a use.
What is the name of it?
2026-03-25 16:01:49.1774454509
What is the name of the expression $(^)(^)^{-1}$?
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If $Y=X^TX$ is invertible, then clearly $$ (X^TX)(X^TX)^{-1}=YY^{-1}=I $$ is the identity matrix.
You might refer instead to $(X^TX)^{-1}X^T$, which is a particular left inverse of $X$, because $$ \bigl((X^TX)^{-1}X^T\bigr)X=(X^TX)^{-1}(X^TX)=I $$ This has a name: it is the Moore-Penrose pseudoinverse of $X$ (it's a very particular case). Proving the properties that show it is indeed the pseudoinverse is easy.