Riv. Mat. Univ. Parma, Vol. 9, No. 2, 2018

Giuseppe Buffoni [a]

On the structure of matrices with positive inverse

Pages: 191-226
Received: 5 July 2018
Accepted in revised form: 24 January 2019
Mathematics Subject Classification (2010): 15B48, 65F30.
Keywords: Monotone matrices, nonnegative and nonpositive perturbations, monotonicity properties of the inverse, preserving monotonicity.
Author address:
[a]: CNR - IMATI, Via Bassini 15, 20133 Milano, Italy

Full Text (PDF)

Abstract: This paper focuses on how monotone+ matrices, i.e., real nonsingular matrices with positive inverse, can either be perturbed or decomposed in such a way that the inverse-positivity is preserved and proved. Let a real matrix \(A\) be split into its components: diagonal entries \(D\), nonpositive \(-B\) and nonnegative \(C\) off-diagonal entries: \(A=D-B+C\). Monotone+ matrices with only two components and their perturbations are identified by investigating the properties of the splittings \(D-B\), \(D+C\) and \(D-B+C\). Monotone+ matrices characterized by three components are identified by means of more involved decompositions of \(A\) or suitable transformations of \(A\), preserving the inverse-positivity, that emphasize the basic properties leading to inverse positivity. Special complex monotone+ matrices are described. The analysis is strongly based on some monotonicity properties of nonpositive and nonnegative perturbations of a monotone+ matrix preserving the inverse-positivity. The results are illustrated by numerical examples.

A. Berman and R. J. Plemmons, Nonnegative matrices in the mathematical science, SIAM, Philadelphia, 1994. MR1298430
A. Berman and R. J. Plemmons, Eight types of matrix monotonicity, Linear Algebra and Appl. 13 (1976), 115–123. MR0395185
D. A. Bini, G. Latouche and B. Meini, Solving matrix polynomial equations arising in queueing problems, Linear Algebra Appl. 340 (2002), 225–244. MR1869430
F. Bouchon, Monotonicity of some perturbations of irreducibly diagonally dominant \(M\)-matrices, Numer. Math. 105 (2007), 591–601. MR2276761
J. H. Bramble and B. E. Hubbard, New monotone type approximations for elliptic problems, Math. Comp. 18 (1964), 349–367. MR0165702
J. H. Bramble and B. E. Hubbard, On a finite difference analogue of an elliptic boundary problem which is neither diagonally dominant nor of non-negative type, J. Math. and Phys. 43 (1964), 117–132. MR0162367
G. Buffoni and A. Galati, Matrici essenzialmente positive con inversa positiva, Boll. Un. Mat. Ital. (4) 10 (1974), 98–103. MR0374165
G. Buffoni, Nonnegative and skew-symmetric perturbations of a matrix with positive inverse, Math. Comp. 54 (1990), 189-194. MR0995208
G. Buffoni, Perturbation of a matrix with positive inverse, Riv. Mat. Univ. Parma (4) 16 (1990), 251–262. MR1105747
L. Collatz, Aufgaben monotoner Art, Arch. Math. 3 (1952), 366–376. MR0053603
J. E. Dennis, Jr., J. F. Traub and R. P. Weber, On the matrix polynomial, lambda-matrix and block eigenvalue problems, Computer Science Technical Reports, Cornell Univ., Ithaca, N.Y., and Carnegie-Mellon Univ., Pittsburgh, PA, 1971. URL
J. E. Dennis, Jr.,J. F. Traub and R. P. Weber, The algebraic theory of matrix polynomials, SIAM J. Numer. Anal. 13 (1976), 831–845. MR0432675
J. E. Dennis, Jr., J. F. Traub and R. P. Weber, Algorithms for solvents of matrix polynomials, SIAM J. Numer. Anal. 15 (1978), 523–533. MR0471278
K. Fan, Topological proofs for certain theorems on matrices with non-negative elements, Monatsh. Math. 62 (1958), 219–237. MR0095856
L. Farina and S. Rinaldi, Positive linear systems: Theory and applications, Wiley-Interscience, New York, 2000. MR1784150
M. Fiedler and V. Pták, On matrices with non-positive off-diagonal elements and positive principal minors, Czechoslovak Math. J. 12 (1962), 382–400. MR0142565
G. Frobenius, über Matrizen aus nicht negativen Elementen, S.-B. Preuss Akad. Wiss. Berlin (1912), 456–477. zbMATH
F. R. Gantmacher, The theory of matrices, Vol. I, Chelsea Publishing Co., New York, 1959. MR0107649
F. R. Gantmacher, The theory of matrices, Vol. II, Chelsea Publishing Co., New York, 1959. MR0107649
A. V. Gavrilov, A sufficient condition for the monotonicity of a positive-definite matrix, Comput. Math. Math. Phys. 41 (2001), 1237–1238. MR1869888
F. Goldberg, On monotonicity-preserving perturbations of \(M\)-matrices, arXiv:1308-0844, preprint, 2013.
R. D. Haynes, M. R. Trummer and S. C. Kennedy, Persistently positive inverses of perturbed \(M\)-matrices, Linear Algebra Appl. 422 (2007), 742–754. MR2305154
J. Huang and T.-Z. Huang, The inverse positivity of perturbed tridiagonal \(M\)-matrices, Linear Algebra Appl. 434 (2011), 131–143. MR2737237
J. Huang, R. D. Haynes and T.-Z. Huang, Monotonicity of perturbed tridiagonal \(M\)-matrices, SIAM J. Matrix Anal. Appl. 33 (2012), 681–700. MR2970225
T. Ikeda, Maximum principle in finite element models for convection-diffusion phenomena, Mathematics Studies, North-Holland, Amsterdam, 1983. MR0683102
W. Kratz and E. Stickel, Numerical solution of matrix polynomial equations by Newton's method, IMA J. Numer. Anal. 7 (1987), 355–369. MR0968530
S. C. Kennedy and R. D. Haynes, Inverse positivity of perturbed tridiagonal \(M\)-matrices, Linear Algebra Appl. 430 (2009), 2312–2323. MR2508297
M. A. Krasnoselskii, Positive solutions of operator equations, Noordhoff, Groningen, 1964. MR0181881
G. I. Marčuk, Metodi del calcolo numerico, Editori Riuniti, Roma, 1984.
T. Netzer and A. Thom, About the solvability of matrix polynomial equations, Bull. Lond. Math. Soc. 49 (2017), 670–675. MR3725487
J. M. Ortega and W. C. Rheinboldt, Monotone iterations for nonlinear equations with application to Gauss-Seidel methods, SIAM J. Numer. Anal. 4 (1967), 171–190. MR0215487
A. Ostrowski, über die determinanten mit überwiegender Hauptdiagonale, Comment. Math. Helv. 10 (1937), 69–96. MR1509568
J. R. Rice, Numerical methods, software and analysis, McGraw-Hill, New York, 1983.
E. E. Tyrtyshnikov, A brief introduction to numerical analysis, Birkhäuser, Boston, MA, 1997. MR1442956
R. S. Varga, Factorization and normalized iterative methods, Boundary problems in differential equations, R. E. Langer, ed., Univ. of Wisconsin Press, Madison, Wis., 1960, 121–142. MR0121977
R. S. Varga, Matrix iterative analysis, Prentice-Hall, Englewood Cliffs, N.J., 1962. MR0158502
T. Vejchodský, Necessary and sufficient condition for the validity of the discrete maximum principle, Proceedings of Algoritmy 2012, 1–10. URL

Home Riv.Mat.Univ.Parma