$\forall C\in \mathbb{R}^{n\times n}, \ \ \ PAP-CMC^T\geq 0\ \ \ \Longrightarrow\ \ \ A-CMC^T\geq 0$

Question

let M be a positive definite $n\times n$ matrix and A a positive semidefinite $n\times n$ matrix and P is an orthogonal projector of some subspace of $\mathbb{R}^n$ into $\mathbb{R}^n$ so is this implication correct $$\forall C\in \mathbb{R}^{n\times n}, \ \ \ PAP-CMC^T\geq 0\ \ \ \Longrightarrow\ \ \ A-CMC^T\geq 0$$

Answer 1

No. For instance, consider $$ P=C=\pmatrix{1&0\\ 0&0},\ A=\pmatrix{1&1\\ 1&1},\ M=I_2. $$ Then $$ PAP-CMC^T=0,\quad A-CMC^T=\pmatrix{0&1\\ 1&1}\not\ge0. $$ The statement is true, however, if $C$ is non-singular. In this case, since $PAP\ge CMC^T>0$, the orthogonal projector $P$ must be non-singular and hence it must be the identity matrix. The implication in question thus reduces to a tautology.