A linear algebraic view of partition regular matrices
نویسندگان
چکیده
1 Rado showed that a rational matrix is partition regular over N if and only if it satisfies the columns 2 condition. We investigate linear algebraic properties of the columns condition, especially for oriented 3 (vertex-arc) incidence matrices of directed graphs and for sign pattern matrices. It is established that 4 the oriented incidence matrix of a directed graph Γ has the columns condition if and only if Γ is strongly 5 connected, and in this case an algorithm is presented to find a partition of the columns of the oriented 6 incidence matrix with the maximum number of cells. It is shown that a sign pattern matrix allows the 7 columns condition if and only if each row is either all zeros or the row has both a + and −. 8 AMS subject classifications: (2010) 15A03, 05C50, 15B35, 05D10. 9
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