## PIMS Distinguished Speaker: Bryan L. Shader

- Date: 10/25/2012
- Time: 15:30

Lecturer(s):
Bryan L. Shader,

*University of Wyoming*
Location:

University of Victoria

Topic:

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

Description:

Abstract:

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.

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