Zeroing Diagonals, Conjugate Hollowization, and Characterizing Nondefinite Operators

TL;DR

This paper proves a conjecture about transforming pairs of traceless matrices into hollow or almost hollow forms via orthogonal similarity, revealing new characterizations of real traceless matrices and extending classical diagonalization results.

Contribution

It introduces a general theorem on zeroing diagonals of matrix pairs, characterizes real traceless matrices, and extends Fillmore's theorem on matrices with constant diagonals.

Findings

Existence of orthogonal V making L hollow and M almost hollow

New characterizations of real traceless matrices

Extended version of Fillmore's theorem on constant diagonals

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.