I'd like to prove the equvialence of the following statements.
For $K\in \mathbb{ C }^{ p\times q }$ :
i) ${ I }_{ p }-{ KK }^{ * }$ is invertible
ii) ${ I }_{ q }-{ K }^{ * }K$ is invertible
iii) $\begin{pmatrix}{ I }_{ p }&K \\{ { K }^{ * } }&{ I }_{ q }\end{pmatrix}$ is invertible
So I'm not really sure if my thoughts are right but for i) to be invertible we need to have that ${ KK }^{ * }$ is invertible. If ${ KK }^{ * }$ is invertible we must have that $K$ and ${K }^{ * }$ are also invertible so trivially ${ K }^{ * }K$ is invertible since its also a product of invertible matrices. And for the third part its now obvious that the given Matrix has a full rank beause of the invertiblity of $K$ and ${K }^{ * }$. Is this reasoning correct?
And a further question if I'd like to get the inverse of the last matrix how do I proceed, I get stuck finding the concrete inverse of i) which I need for the next step in $$\left.\begin{pmatrix}{ I }_{ p }&K\\ { 0 }&{ I }_{ p }-{ KK }^{ * }\end{pmatrix}\middle|\begin{pmatrix}{ K }^{ * }&0\\ 0 & I \end{pmatrix}\right..$$
Appreicate any help to understand this problem