David R. Nicholus | Searcharxiv

arXiv2026

Zeroing Diagonals, Conjugate Hollowization, and Characterizing Nondefinite Operators

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.

math.NA↗

arXiv2023

Antidiagonal Operators, Antidiagonalization, Hollow Quasidiagonalization -- Unitary, Orthogonal, Permutation, and Otherwise - and Symmetric Spectra

After summarizing characteristics of antidiagonal operators, we derive three direct sum decompositions characterizing antidiagonalizable linear operators - the first up to permutation-similarity, the second up to similarity, and the third up to unitary similarity. Each corresponds to a unique quasidiagonalization. We prove the permutation-similarity direct sum decomposition defines a hollow quasidiagonalization of a traceless antidiagonalizable operator and gives the real Schur decomposition of a real antisymmetric antidiagonal operator. We use this to derive an orthogonal antidiagonalization of a general real antisymmetric operator. We prove the similarity direct sum decomposition defines the eigendecomposition of an antidiagonalizable operator that is diagonalizable, and we give a characterization of this eigendecomposition. We show it also defines the Jordan canonical form for a general antidiagonalizable operator. This leads to a further characterization of antidiagonalizable operators in terms of spectral properties and a characterization as the direct sum of traceless $2 \times 2$ matrices with the exception of a single $1 \times 1$ matrix as an additional summand for operators of odd size. We discuss numerous implications of this for properties of the square of an antidiagonalizable operator, a characterization of operators that are both diagonalizable and antidiagonalizable, nilpotency of antidiagonalizable operators, unitary diagonalizations of normal antidiagonal operators, symmetric and antisymmetric antidiagonalizations, and centrosymmetric diagonalizations and antidiagonalizations. Finally, we prove the unitary similarity direct sum decomposition defines the Schur decomposition, as well as a unitary quasidiagonalization, of a unitarily antidiagonalizable operator.

math.RA↗