A linear algebraic view of partition regular matrices

Date
2010-12-01
Authors
Hogben, Leslie
McLeod, Jillian
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Mathematics
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Abstract

Rado showed that a rational matrix is partition regular over N if and only if it satisfies the columns condition. We investigate linear algebraic properties of the columns condition, especially for oriented (vertex-arc) incidence matrices of directed graphs and for sign pattern matrices. It is established that the oriented incidence matrix of a directed graph Γ has the columns condition if and only if Γ is strongly connected, and in this case an algorithm is presented to find a partition of the columns of the oriented incidence matrix with the maximum number of cells. It is shown that a sign pattern matrix allows the columns condition if and only if each row is either all zeros or the row has both a + and −.

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This is a manuscript of an article from Linear Algebra and its Applications 433 (2010): 1809, doi:10.1016/j.laa.2010.06.041. Posted with permission.

Keywords
Columns condition, Partition regular matrix, Oriented incidence matrix, Sign pattern
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