Given $m<n$. Suppose that $H$ and $K$ be $m \times n$ and $n\times (n-m)$ matrices such that rank$(H)=m$, rank$(K)=n-m$, and $HK=0$. For fixed non singular symmetric matrix $A$ define \begin{equation} \ P=(K^TAK)^{-1}K^TA. \end{equation}
I'd like to express $P$ in term of $H$ and $A$ without involving $K$. I am sure we can do this because I could prove $KP=I-A^{-1}H^T(HA^{-1}H^T)^{-1}H$ where $I$ is the identity matrix. However, what I want is expressing $P$.
Can anyone help me? Any help will be appreciated. Thanks
Consider $\widetilde{K} = 2K$. We have:
So, $\widetilde{K}$ satisfies the same conditions as $K$. Now, define
$$\widetilde{P} := (\widetilde{K}^T A \widetilde{K})^{-1} \widetilde{K}^T A.$$
It is easy to see that
$$\widetilde{P} = (\widetilde{K}^T A \widetilde{K})^{-1} \widetilde{K}^T A = ((2K)^T A (2K))^{-1} (2K)^T A = \frac{1}{2} (K^T A K)^{-1} K^T A = \frac{1}{2}P.$$
But the only thing we changed was $K$ (while satisfying the given conditions), so $P$ must depend on $K$, and you cannot get a formula depending only on $H$ and $A$ (neither of which we've changed), except when $P = 0$.