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 Aug 18 revised An argument to prove asymptotic expansions added 748 characters in body Aug 18 revised An argument to prove asymptotic expansions added 129 characters in body Aug 18 revised An argument to prove asymptotic expansions corrected typo Aug 18 asked An argument to prove asymptotic expansions Aug 18 comment Standard terminology for the relation between $A$ and $B$ if $B= Q^t A P$? Yes, thanks for writing up these clarifications. But what these "conjugations" actually mean, was more or less clear to me. I just wanted to know if the case under question has a standard name. I'm trying to write a proof where I have to change a lot of times coordinates in V and W in suitable ways, and I wanted to say: at every step I get a matrix which is "..." to the previous, using a standard term. Maybe i should just write "equivalent" (after defining what I mean). Aug 18 accepted Inequality for a selfadjoint operator on Hilbert space Aug 18 comment Inequality for a selfadjoint operator on Hilbert space I like this point of view very much. Thanks. I think a canonical reference for the spectral theorem you cite could be Theorem VIII.4 in the first Volume of Reed-Simon. Aug 18 comment Standard terminology for the relation between $A$ and $B$ if $B= Q^t A P$? @M Turgeon. Yes, thanks, I forgot to write it. Aug 18 revised Standard terminology for the relation between $A$ and $B$ if $B= Q^t A P$? added 16 characters in body Aug 17 asked Standard terminology for the relation between $A$ and $B$ if $B= Q^t A P$? Aug 17 comment Inequality for singular values ok, I got it: if $u$ is an eigenvector of $A^TA$ then $Au$ is an eigenvector of $AA^T$ with same eigenvalue, and viceversa. Aug 17 comment Inequality for singular values Why $A^T A$ and $A A^T$ have the same eigenvalues? Aug 17 comment Inequality for singular values Thanks for this fast answer. As far as I understand the same reasoning doesn't work if $A,B$ are symmetric and if the $\mu_i$'s denote eigenvalues instead of singular values. Am I wrong? Do in that case the inequalities not hold? Aug 17 revised Inequality for singular values deleted 2 characters in body Aug 17 revised Inequality for a selfadjoint operator on Hilbert space added tag Aug 17 asked Inequality for singular values Aug 17 awarded Teacher Aug 16 revised Inequality for a selfadjoint operator on Hilbert space added tag spectral theory Aug 16 comment Inequality for a selfadjoint operator on Hilbert space @timur: I edited the answer to clarify. Aug 16 revised Inequality for a selfadjoint operator on Hilbert space clarification to firste equality after comments of Byron Schmuland and timur.