ILAS2016 — 11–15 July 2016 — KU Leuven, Belgium

20th Conference of the International Linear Algebra Society (ILAS)

20th ILAS Conference

Contributed talks on Numerical Range and Location of Eigenvalues

Mon 14:30–15:00, Room AV 04.17, Chair: Brian Lins
On the second largest eigenvalue of interval matrices
Milica Andjelic (Kuwait University)

By an interval matrix we assume a real matrix whose all entries lie in some closed interval $[a,b]$ ($-\infty <a < b < +\infty$). In [1] the exact ranges of extremal eigenvalues of real symmetric interval matrices were determined. The paper also contains several open problems. One of them is the following:

Problem: For a given integer $j$ with $2\leq j\leq n-1$ determine $\max\{\lambda_j(A): A\in S_n[a,b]\}$ and $\min\{\lambda_j(A): A\in S_n[a,b]\}$, and also identify which matrices (if any) attain the above extremal values.

We offer the solution to the above problem provided $j=2$ and $j=n-1$ i.e. we determine both lower and upper bounds of the second largest and the second smallest eigenvalue of symmetric interval matrices. All bounds are sharp.

  1. X. Zhan. Extremal eigenvalues of real symmetric matrices with entries in an interval. SIAM J. Matrix Anal. Appl., 27:851-860, 2006.