Zeroing Diagonals, Conjugate Hollowization, and Characterizing Nondefinite Operators

Abstract

We prove the conjecture by Damm and Fassbender that, for real traceless matrices $L,M$, there exists orthogonal $R$ such that $\mathrm{diag}(R^\top L R) = (0,...,0,0,0)$ and $\mathrm{diag}(R M R^\top) = (0,...,0,*,*)$. We also prove for any pair $L,M$ of complex Hermitian traceless matrices, there exists a unitary $U$ such that $\mathrm{diag}(U^* L U) =\mathrm{diag}(U M U^*) = (0,...,0)$. The claims comprise a corollary to our more general theorem for $L,M$ of arbitrary trace. We also discuss severe limitations upon generalizing our theorem to general complex $L,M$. 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 a novel characterization of real traceless matrices and complex Hermitian traceless matrices, strengthening the seminal theorem by Fillmore that every complex square matrix is unitarily similar to a hollow matrix. Our results are contextualized in a characterization of nondefinite matrices as a more general environment for introducing 0s to the main diagonal.