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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 $<A,B> = trace(A^{T}\cdot B).$

Bill chen
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  • In the SVD, $U$ and $V$ are orthogonal matrices, so $UU^T=I$ and $VV^T=I$, where $I$ is the identity matrix of the proper size. Then $P(X)=X$ and your question is trivial, since $\langle P(E_{i,j}), E_{i',j'}\rangle=\operatorname{trace}(E_{i,j}^T E_{i',j'})=\operatorname{trace}(0)=0$. Or am I missing something? Maybe $*$ is not matrix multiplication? – dvdgrgrtt Oct 11 '23 at 05:23

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