Given a matrix $A$ with rank $r$. Suppose its reduced svd is $USV^{T}$. Denote $E_{i,j}$ as matrix with only entry $(i,j)$ equaling to 1 and others all zeros. Denote projection operator $P$ as $$ P(X) = U*U^{T}*X + X*V*V^{T} - U*U^{T}*X*V*V^{T}. $$
How to prove $<P(E_{i,j}),E_{i',j'}> = 0$ if $i\neq i'$ or $j\neq j'$, where $<A,B> = trace(A^{T}\cdot B).$