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 $\mathrm{diag}(R^{\top }LR)=(0,...,0,0,0)$ and $\mathrm{diag}(RM{R}^{\top })=(0,...,0,\ast ,\ast )$. We also prove for any pair $L,M$ of complex Hermitian traceless matrices, there exists a unitary $U$ such that $\mathrm{diag}(U^{\ast }LU)=\mathrm{diag}(UM{U}^{\ast })=(0,...,0)$. 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.

16 pages, 0 figures. This is a significant update to and replacement for the previous submission. The previous submission contained an error. The main Theorem in this version is Theorem 5.4, and Damm and Fassbender's conjecture is proven in Corollary 5.5