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I see from my examples that the product of a positive semidefinite matrix(graph Laplacian) and an indefinite matrix (real matrices), comes to be a positive semidefinite matrix. Is there a proof for such a thing or did it just happened based on my values?

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  • $\begingroup$ This is not true in general, you can go to their spectral decomposition to see it. Suppose they have same eigen vector and eigen value of the first are positive, then taking negative eigen values for the second one gives you a counter example $\endgroup$ – P. Quinton Mar 4 at 7:40
  • $\begingroup$ As another counter example, note that the identity matrix is positive (semi-)definite, and yet its product with an indefinite matrix will be indefinite. I would suppose that what you observed is more an exception than a rule. $\endgroup$ – Florian Mar 4 at 8:50

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