Localization of eigenvalues of Doubly Cyclic Matrices

Abstract

Fix positive numbers $\alpha$ and $\beta$. For the family of doubly cyclic matrices of the form $diag(a_1,a_2,...,a_n)-diag(b_1,b_2,...,b_n){\Sigma }_{\ast }$, where ${\Sigma }_{\ast }$ is a permutation matrix for the $n$-cycle $1\to 2$, $2\to 3$, ... ,$n-1\to n$, $n\to 1$ [cycle notation (1, 2, ... , n-1, n)], and with fixed geometric mean $\alpha$ for the $a_k$'s and $\beta$ for the $b_k$'s, the maximum number of eigenvalues in the left half-plane is attained by $diag(\alpha ,\alpha ,...,\alpha )-diag(\beta ,\beta ,...,\beta ){\Sigma }_{\ast }$. This confirms a conjecture of C. Johnson, Z. Price, and I. Spitkovsky.' Moreover, the complete range of possibilities for the number of eigenvalues in the left half-plane is demonstrated: if $\alpha <\beta$, then any odd number between 1 and the maximum, inclusive, is attainable, and these are the only possibiliites.

34 pages, 2 figures

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