A linear algebra question about projection matrix

Question

Suppose Pr is the projection matrix onto the row space of A. Then how to prove APr=A?

Answer 1

I'm assuming $A$ is a real matrix. Assume $P$ is a projection onto the row space of $A$, i.e. the range of $A^T$. (Projections onto subspaces are not unique, though there is only one orthogonal projection, which might be what you meant.) Then by definition of projection, we have $PA^Tx = A^Tx$ for all $x$. Thus $PA^T = A^T$. Taking transposes gives $AP^T = A$. Using the diagonalization theorem for self adjoint linear maps, it can be shown that a projection $P$ is an orthogonal projection if and only if $P^T = P$. So if your $P$ is an orthogonal projection, then $AP = A$.