I have a problem below:
Given a full rank $m\times n$ matrix $H$ with $m<n$. Let $K$ is an $n \times (n-m)$ matrix in which each column is the element of the basis of the subspace $\{x\in \mathbb{R}^n:Hx=0\}.$ Now define \begin{equation} P=K(K^TAK)^{-1}K^TA. \end{equation} where $A$ is a given non singular symmetric matrix. Can we express $P$ explicitly in term of $H$ and doesn't contain matrix $K$? Thank you in advance. Any help will be appreciated.
Try to consider $$ P=I-Q=I-A^{-1}H^T(HA^{-1}H^T)^{-1}H. $$
Then $$ H(P+Q)=H[A^{-1}H^T(HA^{-1}H^T)^{-1}H+K(K^TAK)^{-1}K^TA]=H $$ and $$ K^TA(P+Q)=K^TA[A^{-1}H^T(HA^{-1}H^T)^{-1}H+K(K^TAK)^{-1}K^TA]=K^TA. $$ Hence, if $[H^T,A^TK]$ is not rank-deficient, $P+Q=I$.