How to decomposition a Hermitian matrix subtracted from Identity?

Question

I've got a complex vector$A\in \mathcal{C}^{M\times 1}$, where$A^H$means the conjugate transpose of $A$, and $A^{-1}$ means the inverse or pseudo-inverse of $A$. The target matrix is $X$.
$X=I-A(A^HA)^{-1}A^H$
Since $X=I-A(A^HA)^{-1}A^H=I-AA^{-1}(AA^{-1})^H=I-PP^H$, where $PP^H$ is a Hermitian matrix, $X\in \mathcal{C}^{M\times M}$ is also a Hermitian matrix.
My problem is how to decomposition $X$ into $X=vv^H$, where $v\in \mathcal{C}^{M\times 1}$. Could you give me the formula of $v$?