Suppose $A$ and $B$ are positive symmetric and semidefinite matrices, and $vAv^{\top} < vBv^{\top}$. What can we say about $A$ and $B$ (in terms of eigenvalues/determinants), knowing nothing about $v$?
2026-03-30 00:24:36.1774830276
What can we say about these matrices in terms of eigenvalues/determinants?
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I'm assuming that $v$ is a specific vector.
All you can say that there is at least one eigenvalue of $B$ that is larger than one eigenvalue of $A$. In other words, the spectrum of $A$ does not dominate that of $B$. I don't think you can say anything more.