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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.

15B48 Positive matrices and their generalizations; cones of matrices
15A09 Theory of matrix inversion and generalized inverses
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