Under what conditions this equality holds?

Question

Consider $P$ and $K$ be two $n\times m$ and $m\times n$ matrices.

under which conditions $$ (I-KP)^{-1} K=K((I-PK)^{-1})$$ holds?

Answer 1

Whenever both inverses $(I-KP)^{-1}$ and $(I-PK)^{-1}$ exist, this equation holds, because $$\matrix{K-KPK & = & K-KPK\\ K(I-PK)& = &(I-KP)K\\ K & = & (I-KP)K\,(I-PK)^{-1}\\ (I-KP)^{-1}K &=& K(I-PK)^{-1}&. }$$

Answer 2

It looks like the above will hold when $ I - KP = I - PK= I$, which will imply that $KP = PK = 0$.
This, in turn, means that $n=m$ for the equality to hold. It also means that the $i^{th}$ row of $K$ must be orthogonal to the $i^{th}$ column of $P$. ($1\leq i \leq m=n$)