Zeroing Diagonals, Conjugate Hollowization, and Characterizing Nondefinite Operators

Abstract

We prove the conjecture by Damm and Fassbender that, for any pair $L,M$ of real traceless matrices, there exists an orthogonal $V$ such that $V^{-1} L \, V$ is hollow and $V M V^{-1}$ is almost hollow, where a matrix is hollow if and only if its main diagonal consists only of 0s, and a traceless matrix is almost hollow if and only if all its main diagonal elements are 0 except, at most, the last two. The claim is a corollary to our considerably more general theorem, as well as another corollary, revealing conditions on $L,M$ under which 0s can be introduced by $V$ to all but the first or first two diagonal elements of $V^{-1} L \, V$ and to all but the last two diagonal elements of $V M V^{-1}$. By setting $L = M$, much is revealed concerning freedom and constraint involved in introducing 0s to the diagonal of a single operator. From this we prove novel characterizations of real traceless matrices, and a stronger version of the seminal theorem by Fillmore that every real matrix is orthogonally similar to a matrix with a constant main diagonal. Our results are contextualized in a characterization and classification of nondefinite matrices by, roughly, how many zeros can be introduced to their diagonals, and it what ways.