Contributed talks on Numerical Range and Location of Eigenvalues
Mon 14:30–15:00, Room AV 04.17, Chair: Brian LinsBy 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.
- X. Zhan. Extremal eigenvalues of real symmetric matrices with entries in an interval. SIAM J. Matrix Anal. Appl., 27:851-860, 2006.