Analysis and Operator Theory Seminar - Charles Baker

February 8, 2018
Thursday, February 15, 2018 - 11:30am to 12:20pm
Cockins Hall 240
math_sculpture

Title: Localization of eigenvalues of doubly cyclic matrices

Speaker: Charles Baker (OSU)

Abstract: For fixed positive $\alpha$ and $\beta$, and a fixed integer $n$, $n \geq 2$, we consider the family of matrices $\operatorname{diag}(a_1, a_2, \dotsc, a_n) - \operatorname{diag}(b_1, b_2, \dotsc, b_n) \Sigma_*$, where all the $a_k$'s and $b_k$'s are positive, the geometric mean of the $a_k$'s and $b_k$'s must be $\alpha$ and $\beta$, respectively, and $\Sigma_*$ denotes the permutation matrix corresponding to the cycle $(1, 2, \dotsc, n)$.

C. Johnson, Z. Price, and I. Spitkovsky conjectured that in this family, the number of eigenvalues in the left half-plane is maximized by $\alpha I - \beta \Sigma_*$; in a joint work with Boris Mityagin, we proved this conjecture. Moreover, we demonstrated the complete range of possibilities for the number of eigenvalues in the left half-plane; if $\alpha < \beta$, then any odd number between 1 and the maximum, inclusive, is attainable.

This talk will briefly remind the audience of the structure of the proof, in particular our choice to only move the extreme elements of the vectors (or multi-sets), and then proceed to flesh out the details.

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