PIMS Distinguished Speaker: Bryan L. Shader

  • Date: 10/25/2012
  • Time: 15:30
Bryan L. Shader, University of Wyoming

University of Victoria


Construction of matrices with a given graph and prescribed interlaced spectral data

Inverse eigenvalue problems have received considerable attention, and arise frequently in engineering applications. Many inverse eigenvalue problems reduce to the construction of a matrix with prescribed spectral data. One interesting problem is based on the Cauchy interlacing inequalities for symmetric matrices, which asserts that if A is a symmetric matrix of order n and B a principal submatrix of order n-1, then the eigenvalues λ1, …, λn  of A interlace the eigenvalues  μ1, …,  μn-1  of B; that is: λ1≤ μ1≤ λ2 ≤... ≤ μn-1≤ λn.  Duarte has proven that when each of the inequalities is strict, for each tree T on n vertices there exists a real symmetric matrix A whose graph is T such that A has eigenvalues λ1, …, λn and the principal submatrix obtained from A by deleting its last row and column has eigenvalues μ1, …,  μn-1.  In this talk we will show how we can combine some basic analysis (the Implicit Function Theorem), some basic linear algebra (the structure of the centralizer of a symmetric matrix) to extend Duarte’s result to arbitrary connected graphs.

This is joint work with Keivan Hassani Monfared.
Other Information: 

Location:  DSB C128