Vol. 9, No. 2, 2016

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The kernel of the matrix $\lbrack i\mskip-2mu j \pmod n\rbrack$ when $n$ is prime

Maria I. Bueno, Susana Furtado, Jennifer Karkoska, Kyanne Mayfield, Robert Samalis and Adam Telatovich

Vol. 9 (2016), No. 2, 265–280
Abstract

In this paper, we consider the n × n matrix whose (i,j)-th entry is ij(modn) and compute its rank and a basis for its kernel (viewed as a matrix over the real numbers) when n is prime. We also give a conjecture on the rank of this matrix when n is not prime and give a set of vectors in its kernel, which is a basis if the conjecture is true. Finally, we include an application of this problem to number theory.

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Keywords
rank of a matrix, kernel of a matrix, bisymmetric matrix.
Mathematical Subject Classification 2010
Primary: 15A03, 11M06, 11M20
Milestones
Received: 14 October 2014
Revised: 13 April 2015
Accepted: 16 April 2015
Published: 2 March 2016

Communicated by Kenneth S. Berenhaut
Authors
Maria I. Bueno
Mathematics Department and College of Creative Studies
The University of California, Santa Barbara
Santa Barbara, CA 93106-3080
United States
Susana Furtado
CEAFEL
Faculdade de Economia
Universidade do Porto
4200-464 Porto
Portugal
Jennifer Karkoska
Applied Mathematics Department
Rensselaer Polytechnic Institute
Troy, NY 12180
United States
Kyanne Mayfield
Fast Enterprises LLC
Madison, WI 53704
United States
Robert Samalis
Department of Mathematics
University of Georgia
Athens, GA 30602
United States
Adam Telatovich
Department of Mathematics
Pennsylvania State University
State College, PA 16803
United States