On the eigenvalue-separation properties of real tridiagonal matrices

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Constructive Mathematical Analysis
Volume:6 Issue:4
On the eigenvalue-separation properties of real tridiagonal matrices

On the eigenvalue-separation properties of real tridiagonal matrices

Authors : Yan Wu, Ludwig Kohaupt

Pages : 210-221

Doi:10.33205/cma.1330647

View : 131 | Download : 135

Publication Date : 2023-12-15

Article Type : Research Paper

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

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