Cofactors and eigenvectors of banded Toeplitz matrices: Trench formulas via skew Schur polynomials

Abstract

The Jacobi-Trudi formulas imply that the minors of the banded Toeplitz matrices can be written as certain skew Schur polynomials. In 2012, Alexandersson expressed the corresponding skew partitions in terms of the indices of the struck-out rows and columns. In the present paper, we develop the same idea and obtain some new applications. First, we prove a slight generalization and modification of Alexandersson's formula. Then, we deduce corollaries about the cofactors and eigenvectors of banded Toeplitz matrices, and give new simple proofs to the corresponding formulas published by Trench in 1985.

23 pages; corrected typos; corrected minor errors in the bibliography

Cited by in corpus (4)

• Eigenvalues of tridiagonal Hermitian Toeplitz matrices with perturbations in the off-diagonal corners
• Localization of eigenvectors of non-Hermitian banded noisy Toeplitz matrices
• Complete homogeneous symmetric polynomials with repeating variables
• Bialternant formula for Schur polynomials with repeating variables