On theLU factorization ofM-matrices
In this paper, we give in Theorem 1 a characterization, based on graph theory, of when anM-matrixA admits anLU factorization intoM-matrices, whereL is a nonsingular lower triangularM-matrix andU is an upper triangularM-matrix. This result generalizes earlier factorization results of Fiedler and Pták (1962) and Kuo (1977). As a consequence of Theorem 1, we show in Theorem 3 that the conditionx T A≧0 T for somex>0, for anM-matrixA, is both necessary and sufficient forPAP T to admit such anLU factorization for everyn×n permutation matrixP. This latter result extends recent work of Funderlic and Plemmons (1981). Finally, Theorem 1 is extended in Theorem 5 to give a characterization, similarly based on graph theory, of when anM-matrixA admits anLU factorization intoM-matrices.
Subject ClassificationsAMS(MOS) 15 A 23 CR: 5.14
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- 1.Berman, A., Plemmons, R.J.: Nonnegative matrices in the mathematical sciences New York: Academic Press, 1979Google Scholar
- 2.Berman, A., Varga, R.S., Ward, R.C. ALPS: Matrices with nonpositive off-diagonal entries. Linear Algebra and Appl.21, 233–244 (1978)Google Scholar
- 3.Fan, Ky: Note onM-matrices. Quart. J. Math. Oxford Ser. (2),11, 43–49 (1960)Google Scholar
- 4.Fiedler, M., Pták, V.: On matrices with nonpositive off-diagonal elements and positive principal minors. Czech. Math. J.12, 382–400 (1962)Google Scholar
- 5.Funderlic, R.E., Mankin, J.B.: Solution of homogeneous systems of linear equations arising from compartmental models. SIAM J. Sci. Statist. Comput. (in press, 1981)Google Scholar
- 6.Funderlic, R.E., Plemmons, R.J.:LU decompositions ofM-matrices by elimination without pivoting. Linear Algebra and Appl. (in press, 1981)Google Scholar
- 7.Kuo, I-wen: A note on factorization of singularM-matrices. Linear Algebra and Appl.16, 217–220Google Scholar
- 8.Rothblum, U.G.: A rank characterization of the number of final classes of a nonnegative matrix. Linear Algebra and Appl.23, 65–68 (1979)Google Scholar
- 9.Varga, R.S.: Matrix iterative analysis. Englewood Cliffs, New Jersey: Prentice-Hall 1962Google Scholar