# Show that ABA=0 if and only if B is of the form C-PCP for some symmetric matrix C

If B is a symmetric matrix such that ABA is well-defined, then show that ABA=0 if and only if B is of the form C-PACPA for some symmetric matrix C.

PA here is the projection matrix of A.

• is this homework? – Dayne Oct 10 '20 at 12:23
• Yes, it is part of my homework. The teacher did not discuss this clearly, so I couldn't quite get a grasp of it. I understood the concept of projection matrices – user530299 Oct 10 '20 at 12:29
• but the part "iff B is of the form C-PCP" just confuses me. – user530299 Oct 10 '20 at 12:30
• This assertion is not generally true. Consider the case $$B=\pmatrix{1&0\\0&1};\quad A=\pmatrix{0&1\\0&0}.$$ I suspect you intend to refer to $ABA^\prime$ and, for $P_A$ to be meaningful, you either need to assume $A$ also is symmetric or else replace $A$ by $(A+A^\prime)/2$ everywhere. – whuber Oct 10 '20 at 14:38