Positive bidiagonal factorization of tetradiagonal Hessenberg matrices

TL;DR

This paper establishes conditions under which oscillatory tetradiagonal Hessenberg matrices can be positively bidiagonally factored, extending spectral theory and analyzing Toeplitz matrices as a case study.

Contribution

It provides new criteria in terms of continued fractions for positive bidiagonal factorization of tetradiagonal Hessenberg matrices and explores their organization in rays.

Findings

Conditions in terms of continued fractions are derived.

Oscillatory Toeplitz matrices admit positive bidiagonal factorization.

Matrices are organized in rays with specific factorization properties.

Abstract

Recently a spectral Favard theorem for bounded banded lower Hessenberg matrices that admit a positive bidiagonal factorization was presented. In this paper conditions, in terms of continued fractions, for an oscillatory tetradiagonal Hessenberg matrix to have such positive bidiagonal factorization are found. Oscillatory tetradiagonal Toeplitz matrices are taken as a case study of matrix that admits a positive bidiagonal factorization. Moreover, it is proved that oscillatory banded Hessenberg matrices are organized in rays, with the origin of the ray not having the positive bidiagonal factorization and all the interior points of the ray having such positive bidiagonal factorization.