# On a criterion of D-stability for P-matrices

Special Matrices (2016)

- Volume: 4, Issue: 1, page 181-188
- ISSN: 2300-7451

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topOlga Y. Kushel. "On a criterion of D-stability for P-matrices." Special Matrices 4.1 (2016): 181-188. <http://eudml.org/doc/277084>.

@article{OlgaY2016,

abstract = {In this paper, we study positive stability and D-stability of P-matrices.We introduce the property of Dθ-stability, i.e., the stability with respect to a given order θ. For an n × n P-matrix A, we prove a new criterion of D-stability and Dθ-stability, based on the properties of matrix scalings.},

author = {Olga Y. Kushel},

journal = {Special Matrices},

keywords = {P-matrices; Q2-matrices; D-stability; matrix scalings; P-matrix powers; stabilization; -matrices; -matrices; -stability; -matrix powers},

language = {eng},

number = {1},

pages = {181-188},

title = {On a criterion of D-stability for P-matrices},

url = {http://eudml.org/doc/277084},

volume = {4},

year = {2016},

}

TY - JOUR

AU - Olga Y. Kushel

TI - On a criterion of D-stability for P-matrices

JO - Special Matrices

PY - 2016

VL - 4

IS - 1

SP - 181

EP - 188

AB - In this paper, we study positive stability and D-stability of P-matrices.We introduce the property of Dθ-stability, i.e., the stability with respect to a given order θ. For an n × n P-matrix A, we prove a new criterion of D-stability and Dθ-stability, based on the properties of matrix scalings.

LA - eng

KW - P-matrices; Q2-matrices; D-stability; matrix scalings; P-matrix powers; stabilization; -matrices; -matrices; -stability; -matrix powers

UR - http://eudml.org/doc/277084

ER -

## References

top- [1] K.J. Arrow, M. McManus, A note on dynamical stability, Econometrica 26 (1958), 448-454. [Crossref] Zbl0107.37201
- [2] C.S. Ballantine, Stabilization by a diagonal matrix, Proc. Amer. Math. Soc., 25 (1970), pp. 728–734. [Crossref] Zbl0228.15001
- [3] A. Berman, R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979. Zbl0484.15016
- [4] B. Cain, Real, 3 × 3, D-stable matrices, J. Res. Nat. Bur. Standards Sect. B, 80B (1976), 75–77. Zbl0341.15009
- [5] D. Carlson, A class of positive stable matrices, J. Res. Nat. Bur. Standards Sect. B, 78B (1974), pp. 1–2. Zbl0281.15020
- [6] M. Fiedler and V. Pták, On matrices with non-positive off-diagonal elements and positive principal minors, Czech. Math. J., 22 (87) (1962), pp. 382–400. Zbl0131.24806
- [7] M.E. Fisher and A.T. Fuller, On the stabilization of matrices and the convergence of linear iterative processes, Proc. Cambridge Philos. Soc., 54 (1958), pp. 417–425. Zbl0085.33102
- [8] I.M. Glazman and Yu.I. Liubich, Finite-Dimensional Linear Analysis: A Systematic Presentation in Problem Form, MIT Press, 1974.
- [9] D.A. Grundy, C.R. Johnson, D.D. Olesky and P. van den Driessche, Products of M-matrices and nested sequences of principal minors, ELA, 16 (2007), pp. 380–388. Zbl1153.15026
- [10] D. Hershkowitz, On the spectra of matrices having nonnegative sums of principal minors, Linear Algebra Appl., 55 (1983), pp. 81–86. [Crossref] Zbl0521.15004
- [11] D. Hershkowitz and C.R. Johnson, Spectra of matrices with P-matrix powers, Linear Algebra Appl., 80 (1986), pp. 159–171. [Crossref] Zbl0612.15013
- [12] D. Hershkowitz and N. Keller, Positivity of principal minors, sign symmetry and stability, Linear Algebra Appl., 364 (2003), pp. 105–124. [Crossref] Zbl1044.15012
- [13] G.V. Kanovei and D.O. Logofet, D-stability of 4-by-4 matrices, Comput. Math. Math. Phys., 38 (1998), pp. 1369–1374. Zbl0967.65053
- [14] D.O. Logofet, Stronger-than-Lyapunov notions of matrix stability, or how"flowers" help solve problems in mathematical ecology, Linear Algebra Appl. 398 (2005), 75-100. Zbl1062.92072
- [15] C.R. Johnson, Sufficient conditions for D-stability, Journal of Economic Theory 9 (1974), 53-62. [Crossref]
- [16] A. Pinkus, Totally positive matrices, Cambridge University Press, 2010.
- [17] J.P. Quirk, R. Ruppert, Qualitative economics and the stability of equilibrium, Rev. Econom. Studies 32 (1965), 311-326. [Crossref]
- [18] A.K. Tang, A. Simsek, A. Ozdaglar and D. Acemoglu, On the stability of P-matrices, Linear Algebra Appl., 426 (2007), 22–32. [Crossref] Zbl1140.15022
- [19] M. Tsatsomeros, Generating and detecting matrices with positive principal minors, Asian Information-Science-Life, 1 (2002), p. 115–132.

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