A trace bound for positive definite connected integer symmetric matrices

James McKee, Pavlo Yatsyna

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Abstract

Let A be a connected integer symmetric matrix, i.e., A=(a_ij)∈Mn(Z) for some n, A=A^T, and the underlying graph (vertices corresponding to rows, with vertex i joined to vertex j if a_ij≠0) is connected. We show that if all the eigenvalues of A are strictly positive, then tr(A)>=2n−1.

There are two striking corollaries. First, the analogue of the Schur–Siegel–Smyth trace problem is solved for characteristic polynomials of connected integer symmetric matrices. Second, we find new examples of totally real, separable, irreducible, monic integer polynomials that are not minimal polynomials of integer symmetric matrices.

Original languageEnglish
Pages (from-to)227-230
Number of pages4
JournalLinear Algebra and Its Applications
Volume444
DOIs
Publication statusPublished - 1 Mar 2014

Keywords

  • Integer symmetric matrices
  • trace problem
  • minimal polynomial

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