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Inverses of Perron complements of inverse $$M$$-matrices. (English) Zbl 0959.15023
The concept of the Perron complement of a nonnegative and irreducible matrix was introduced by C. D. Meyer [Linear Algebra Appl. 114/115, 69-94 (1989; Zbl 0673.15006) and SIAM Rev. 31, No. 2, 240-272 (1989; Zbl 0685.65129)]. In the present paper the author studies the properties of the Perron complement of a matrix which is an inverse of an irreducible $$M$$-matrix. He shows that such complement is again an inverse $$M$$-matrix. He also investigates the directed graph of the inverses of the extended Perron complements (associated to an inverse of an irreducible $$M$$-matrix), and shows that the Perron complement of an inverse of an irreducible tridiagonal $$M$$-matrix is again an inverse of an irreducible tridiagonal $$M$$-matrix and hence totally nonnegative.

##### MSC:
 15B48 Positive matrices and their generalizations; cones of matrices 15A09 Theory of matrix inversion and generalized inverses
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##### References:
 [1] A. Berman, R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, PA, 1994 · Zbl 0815.15016 [2] A. Björck, Numerical Methods for Least Squares Problems, SIAM, Philadelphia, PA, 1966 [3] Crabtree, D.E., Applications of M-matrices to nonnegative matrices, Duke math. J., 33, 197-208, (1966) · Zbl 0142.27101 [4] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, New York, 1985 · Zbl 0576.15001 [5] G. Golub, C. Van Loan, Matrix Computation, third ed., The John Hopkins University Press, Baltimore, MD, 1996 · Zbl 0865.65009 [6] C.R. Johnson, C. Xenophotos, Irreducibility and primitivity of Perron complements: applications of the compressed graph, in: Graph Theory and Sparse Matrix Computations, IMA Vol. Math. Appl., vol. 56, 1993, pp. 101-106 [7] M. Lewin, Totally nonnegative, M-, and Jacobi matrices, SIAM J. Algebraic Discrete Methods 4 (1980) 419-421 [8] Lewin, M.; Neumann, M., On the inverse M-matrix problem for $$(0,1)$$-matrices, Linear algebra appl., 30, 41-50, (1980) · Zbl 0434.05051 [9] Meyer, C.D., Uncoupling the Perron eigenvector problem, Linear algebra appl., 114/115, 69-94, (1989) · Zbl 0673.15006 [10] Meyer, C.D., Stochastic complementation, uncoupling Markov chains, and the theory of nearly reducible systems, SIAM rev., 31, 240-272, (1989) · Zbl 0685.65129 [11] Schneider, H., Theorems on M-splittings of a singular M-matrix which depend on graph structure, Linear algebra appl., 58, 407-424, (1984) · Zbl 0561.65020
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