Spectra of Tridiagonal Matrices over a Field

TL;DR

This paper explores the spectral properties of irreducible tridiagonal matrices over arbitrary fields, providing explicit conditions that generalize known results from real symmetric and Hermitian cases to more general settings.

Contribution

It extends the understanding of spectra of tridiagonal matrices beyond classical cases, offering explicit spectral conditions over arbitrary fields.

Findings

Real symmetric and Hermitian cases require distinct eigenvalues with interlacing spectra.

General fields allow more flexible spectral configurations, with less restrictive conditions.

Explicit spectral conditions are derived for matrices over arbitrary fields.

Abstract

We consider spectra of $n$-by-$n$ irreducible tridiagonal matrices over a field and of their $n-1$-by-$n-1$ trailing principal submatrices. The real symmetric and complex Hermitian cases have been fully understood: it is necessary and sufficient that the necessarily real eigenvalues are distinct and those of the principal submatrix strictly interlace. So this case is very restrictive. By contrast, for a general field, the requirements on the two spectra are much less restrictive. In particular, in the real or complex case, the $n$-by-$n$ characteristic polynomial is arbitrary (so that the algebraic multiplicities may be anything in place of all 1's in the classical cases) and that of the principal submatrix is the complement of a lower dimensional algebraic set (and so relatively free). Explicit conditions are given.