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Permanent of the product of doubly stochastic matrices

  • R. A. Brualdi (a1)

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If A = [aij] is an n × n matrix, the permanent of A is the scalar valued function of A defined by

where the summation extends over all permutations (i1, i2, …, in) of the integers 1, 2, …, n. If we assume that A is a non-negative matrix (that is, that A has non-negative entries) then we come upon an extremely interesting situation. Much work has been done in finding significant upper bounds for the permanent and permanental minors of A (see, e.g. (2, 4–7) in (5) an excellent bibliography is given). If B = [bij] is another n × n non-negative matrix, then we may form the product AB and consider the permanent of this matrix. Unlike the determinant, the permanent is not a multiplicative function. In our circumstances here, however, it is easy to verify and indeed a proof was written down in (1) that

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References

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(1)Brualdi, R. A.Permanent of the direct product of matrices. Pacific J. Math. 16 (1966), 471482.
(2)Brualdi, R. A. & Newman, M.Inequalities for permanents and permanental minors. Proc. Cambridge Philos. Soc. 61 (1965), 741746.
(3)Dulmage, A. L. & Mendelsohn, N. S.Coverings of bipartite graphs. Canad. J. Math. 10 (1958), 517534.
(4)Jurkat, W. B. & Ryser, H. J.Matrix factorizations of determinants and permanents. J. Algebra 3 (1966), 127.
(5)Marcus, M. & Minc, H. Permanents.Amer. Math. Monthly 72 (1965), 577591.
(6)Marcus, M. & Minc, H.A survey of matrix theory and matrix inequalities (Allyn and Bacon; Boston, 1964).
(7)Marcus, M., Minc, H. & Newman, M.Inequalities for the permanent function. Ann. of Math. 75 (1962), 4762.

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