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Nonnegative matrices whose inverses are $ M$-matrices

Author: Thomas L. Markham
Journal: Proc. Amer. Math. Soc. 36 (1972), 326-330
MSC: Primary 15A48
MathSciNet review: 0309970
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Abstract: A characterization of a class of totally nonnegative matrices whose inverses are M-matrices is given. It is then shown that if A is nonnegative of order n and $ {A^{ - 1}}$ is an M-matrix, then the almost principal minors of A of all orders are nonnegative.

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  • [1] M. Fiedler and V. Pták, On matrices with non-positive off-diagonal elements and positive principal minors, Czechoslovak Math. J. 12 (87) (1962), 382-400. MR 26 #134. MR 0142565 (26:134)
  • [2] F. R. Gantmacher and M. G. Kreĭn, Sur les matrices complètement non-negatives et oscillatories, Compositio Math. 4 (1937), 445-476. MR 1556987
  • [3] T. L. Markham, On oscillatory matrices, Linear Algebra and Appl. 3 (1970), 143-156. MR 41 #5392. MR 0260769 (41:5392)

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Keywords: Nonnegative matrix, totally nonnegative matrix, M-matrix, almost principal minor
Article copyright: © Copyright 1972 American Mathematical Society

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