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Let $A$ be an $n \times n$ positive semi-definite matrix, and let $B$ be an $m \times n$ matrix.

Can one say anything relating the eigenvalues of $M_1 := BAB^\top$ and $M_2 := BA^2 B^\top$? In particular, I am interested if a bound on the maximum eigenvalue of one matrix implies an analogous bound for the other matrix.

My elementary observations:

  • Both matrices are positive semi-definite.
  • If $B^\top B = I$, then $M_2 = M_1^2$, so the eigenvalues are directly related. But I am interested in what happens when $B$ does not have orthonormal columns.
  • By diagonalizing $A$, it suffices to prove the claim for diagonal $A$.

My hunch is that if $B^\top B$ were badly behaved in some way, the eigenvalues for $M_2$ can be made to be arbitrarily large/small, but I am not able to formalize this guess.

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