A trace bound for positive definite connected integer symmetric matrices

作者：Mckee James^*; Yatsyna Pavlo

来源：Linear Algebra and Its Applications, 2014, 444: 227-230.

DOI：10.1016/j.laa.2013.11.032

• Integer symmetric matrices
• Trace problem
• Minimal polynomials

Let A be a connected integer symmetric matrix, i.e., A = (a(ij)) is an element of 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) not equal 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.

• 出版日期2014-3-1