IJPAM: Volume 81, No. 2 (2012)

EXTENDING THE HALL-PORSCHING BOUNDS
FOR THE PERRON ROOT

Jorma K. Merikoski
School of Information Sciences
FI-33014, University of Tampere
FINLAND


Abstract. Let $\mathbf{A}\in\mathbb{R}^{n\times n}$ be nonnegative with Perron root $r$ and row sums $s_1,\dots,s_n$, and denote $S=\max_is_i$, $s=\min_is_i$. We improve the Frobenius bounds $s\le r\le S$ by applying them to $\mathbf{DAD}^{-1}$, where $\mathbf{D}$ is obtained from the identity matrix $\mathbf{I}$ by replacing its certain diagonal entries with a suitably chosen positive number. As a special case, in changing only one entry, we obtain the Hall-Porsching bounds.

Received: March 29, 2011

AMS Subject Classification: 15A18, 15B48

Key Words and Phrases: eigenvalue bounds, Perron root, nonnegative matrices

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 81
Issue: 2