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Term Rank of the Direct Product of Matrices

  • Richard A. Brualdi (a1)

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Let A = [aij] be a matrix of 0's and 1's or a (0, 1)-matrix of size m by m′. The term rank of A is denned as the maximal number of 1's of A with no two of the 1's on the same row or colunn. A theorem due to D. König (3, Theorem 5.1, p. 55) asserts that the term rank of A is also equal to the minimal number of rows and columns of A that collectively contain all the 1's. The term rank of A will be denoted by ρ(A). Obviously it is invariant under arbitrary permutations of the rows and columns of A. We assume without loss of generality that all matrices considered have no rows or columns consisting entirely of 0's.

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References

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1. MacDuffee, C. C., The theory of matrices, rev. ed. (New York, 1946).
2. Ore, O., Graphs and matching theorems, Duke Math. J., 22 (1955), 625639.
3. Ryser, H. J., Combinatorial mathematics, Carus Math. Monograph, no. 14 (New York, 1963).
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