A contribution to the connections between Fibonacci numbers and matrix theory

Farber, Miriam; Berman, Abraham

doi:10.2140/involve.2015.8.491

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A contribution to the connections between Fibonacci numbers and matrix theory

Miriam Farber and Abraham Berman

Vol. 8 (2015), No. 3, 491–501

DOI: 10.2140/involve.2015.8.491

Abstract

We present a lovely connection between the Fibonacci numbers and the sums of inverses of $(0, 1)$ -triangular matrices, namely, a number $S$ is the sum of the entries of the inverse of an $n \times n$ $(0, 1)$ -triangular matrix (for $n \geq 3$ ) if and only if $S$ is an integer between $2 - F_{n - 1}$ and $2 + F_{n - 1}$ . Corollaries include Fibonacci identities and a Fibonacci-type result on determinants of a special family of $(1, 2)$ -matrices.

Keywords

Fibonacci numbers, Hessenberg matrix, sum of entries

Mathematical Subject Classification 2010

Primary: 15A15, 11B39, 15A09, 15B99

Milestones

Received: 19 August 2013

Revised: 28 October 2013

Accepted: 5 November 2013

Published: 5 June 2015

Communicated by Robert J. Plemmons

Authors

Miriam Farber
Department of Mathematics Technion–Israel Institute of Technology Technion City, Haifa 3200003 Israel

Abraham Berman
Department of Mathematics Technion–Israel Institute of Technology Technion City, Haifa 3200003 Israel

Vol. 8, No. 3, 2015