On the eigenvalue-separation properties of real tridiagonal matrices

Abstract

In this paper, we give a simple sufficient condition for the eigenvalue-separation properties of real tridiagonal matrices T. This result is much more than the statement that the pertinent eigenvalues are distinct. Its derivation is based on recurrence formulae satisfied by the polynomials made up by the minors of the characteristic polynomial det(xE-T) that are proven to form a Sturm sequence. This is a new result, and it proves the simple spectrum property of a symmetric tridiagonal matrix studied in Grünbaum's paper. Two numerical examples underpin the theoretical findings. The style of the paper is expository in order to address a large readership.

Keywords

• Characteristic polynomial
• Distinct eigenvalues
• Eigenvalue-separation properties
• Minors of determinant
• Sturm sequence
• Tridiagonal matrix

References

1. G. H. Goloub, Ch. F. van Loan: Matrix Computations, The Johns Hopkins University Press, Baltimore and London (1989).
2. F. A. Grünbaum: Toeplitz Matrices Commuting with Tridiagonal Matrices, Linear Algebra Appl., 40 (1981), 25–36.
3. M. Hanke-Bourgeois: Grundlagen der Numerischen Mathematik und des Wissenschaftlichen Rechnens (Foundations of Numerical Analysis and Scientific Computing), B. G. Teubner, Stuttgart, Leipzig, Wiesbaden (2002).
4. F. Stummel, K. Hainer: Introduction to Numerical Analysis, (English Translation by E.R. Dawson of the First Edition of the German Original of 1971,) Scottish Academic Press, Edinburgh (1980).