Tropical bounds for eigenvalues of matrices

Abstract

We show that for all k = 1,...,n the absolute value of the product of the k largest eigenvalues of an n-by-n matrix A is bounded from above by the product of the k largest tropical eigenvalues of the matrix |A| (entrywise absolute value), up to a combinatorial constant depending only on k and on the pattern of the matrix. This generalizes an inequality by Friedland (1986), corresponding to the special case k = 1.

17 pages, 1 figure

References in corpus (4)

• Tropical Scaling of Polynomial Matrices
• Min-plus methods in eigenvalue perturbation theory and generalised Lidskii-Vishik-Ljusternik theorem
• Perturbation of eigenvalues of matrix pencils and optimal assignment problem
• Log-majorization of the moduli of the eigenvalues of a matrix polynomial by tropical roots

Cited by in corpus (5)

• Log-majorization of the moduli of the eigenvalues of a matrix polynomial by tropical roots
• Non-archimedean valuations of eigenvalues of matrix polynomials
• Tropical compound matrix identities
• The Origin and the Resolution of Nonuniqueness in Linear Rational Expectations
• Perturbation of Perron roots and The max-plus spectral theorem