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Does the graph laplacian matrix $L$ form a reproducing kernel- given that the matrix is positive semi-definite. I was told in a hallway by a post doc- a month ago that the pseudo-inverse of $L$ forms a reproducing kernel. What about $L$? I ask because, a kernel is p.d if and only if it is a reproducing kernel. Does the converse hold true?

Even better if someone can putforth:

i) The reasoning behind, why it holds true or not. As, I thought any matrix $G$ formed by a kernel $k(.)$ which can be decomposed as $XX^T$ for some $X$ can be considered a reproducing kernel. Is that not-true?

ii) What are the properties of the metric, $ρ(.)$ induced by a kernel as $ρ(xi,xj)=k(xi,xi)+k(xj,xj)−2k(xi,xj)$ if the kernel is a reproducing kernel? i.e, are there distinguishing properties of a reproducing kernel induced metric that do not hold true for a kernel that is not a reproducing kernel?

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