zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Recent directions in matrix stability. (English) Zbl 0759.15010
This is mainly a survey paper on matrix stability. One starts with classical stability criteria. Then one studies the present sufficient conditions for stability (with particular emphasis on $P$-matrices), the $D$-stability, the additive $D$-stability, and the Lyapunov diagonal stability. One discusses the weak principal submatrix rank property, shared by Lyapunov diagonally semistable matrices, the uniqueness of Lyapunov scaling factors, maximal Lyapunov scaling factors, cones of real positive semidefinite matrices and their applications to matrix stability, and inertia preserving matrices. In this context one derives some original results on stable scaling of complex matrices and on inertia preserving matrices in the acyclic case.
Reviewer: M.Voicu (Iaşi)

15A42Inequalities involving eigenvalues and eigenvectors
93E20Optimal stochastic control (systems)
15B48Positive matrices and their generalizations; cones of matrices
15A12Conditioning of matrices
15-02Research monographs (linear algebra)
Full Text: DOI
[1] Araki, M.: Applications of M-matrices to the stability problems of composite dynamical systems. J. math. Anal. appl. 52, 309-321 (1975) · Zbl 0324.34045
[2] Ballantine, C. S.: Stabilization by a diagonal matrix. Proc. amer. Math. soc. 25, 728-734 (1970) · Zbl 0228.15001
[3] Barker, G. P.; Berman, A.; Plemmons, R. J.: Positive diagonal solutions to the Lyapunov equations. Linear and multilinear algebra 5, 249-256 (1978) · Zbl 0385.15006
[4] Berman, A.; Hershkowitz, D.: Matrix diagonal stability and its implications. SIAM J. Algebraic discrete methods 4, 377-382 (1983) · Zbl 0547.15009
[5] Berman, A.; Hershkowitz, D.: Characterization of acyclic D-stable matrices. Linear algebra appl. 58, 17-31 (1984) · Zbl 0543.15015
[6] Berman, A.; Hershkowitz, D.: Graph theoretical methods in studying stability. Contemp. math. 47, 1-6 (1985) · Zbl 0583.15010
[7] A. Berman and D. Shasha, Inertia preserving matrices, SIAM J. Matrix Anal. Appl., to appear. · Zbl 0813.15015
[8] Berman, A.; Varga, R. S.; Ward, R. C.: ALPS: matrices with nonpositive off-diagonal entries. Linear algebra appl. 21, 233-244 (1978) · Zbl 0401.15018
[9] Berman, A.; Ward, R. C.: ALPS: classes of stable and semipositive matrices. Linear algebra appl. 21, 163-174 (1978) · Zbl 0386.15015
[10] Cain, B. E.: Real 3X3 D-stable matrices. J. res. Nat. bur. Standards sect. B 80, 75-77 (1976) · Zbl 0341.15009
[11] Carlson, D.: Weakly sign-symmetric matrices and some determinantal inequalities. Colloq. math. 17, 123-129 (1967) · Zbl 0147.27502
[12] Carlson, D.: A class of positive stable matrices. J. res. Nat. bur. Standards sect. B 78, 1-2 (1974) · Zbl 0281.15020
[13] Carlson, D.; Datta, B. N.; Johnson, C. R.: A semidefinite Lyapunov theorem and the characterization of tridiagonal D-stable matrices. SIAM J. Algebraic discrete methods 3, 293-304 (1982) · Zbl 0541.15008
[14] Cauchy, A.: Sur l’équation á l’aide de laquelle on détermine LES inégalités seculaires de mouvement des planétes. Oeuv. comp. 9, No. 2, 174-195 (1829)
[15] Cross, G. W.: Three types of matrix stability. Linear algebra appl. 20, 253-263 (1978) · Zbl 0376.15007
[16] Engel, G. M.; Schneider, H.: The Hadamard-fischer inequality for a class of matrices defined by eigenvalue monotonicity. Linear and multilinear algebra 4, 155-176 (1976)
[17] Fisher, M. E.; Fuller, A. T.: On the stabilization of matrices and the convergence of linear iterative processes. Proc. Cambridge philos. Soc. 54, 417-425 (1958) · Zbl 0085.33102
[18] Friedland, S.: Weak interlacing property of totally positive matrices. Linear algebra appl. 71, 95-100 (1985) · Zbl 0609.15010
[19] Fujiwara, M.: On algebraic equations whose roots Lie in a circle or in a half plane. Math. Z. 24, 161-169 (1926)
[20] Gantmacher, F. R.: The theory of matrices. (1960) · Zbl 0088.25103
[21] Goh, B. S.: Global stability in two species interactions. J. math. Biol. 3, 313-318 (1976) · Zbl 0362.92013
[22] Goh, B. S.: Global stability in many species systems. Amer. nat. 111, 135-143 (1977)
[23] Hadeler, K. P.: Nonlinear diffusion equations in biology. Springer lecture notes (1976) · Zbl 0354.65058
[24] Hartfiel, D. J.: Concerning the interior of the D-stable matrices. Linear algebra appl. 30, 201-207 (1980) · Zbl 0442.15003
[25] Hermite, C.: Sur le nombre des racines d’une équation algébrique comprise entre des limites données. J. reine angew. Math. 52, 39-51 (1856) · Zbl 052.1365cj
[26] Hershkowitz, D.: Stability of acyclic matrices. Linear algebra appl. 73, 157-169 (1986) · Zbl 0588.15012
[27] Hershkowitz, D.: Lyapunov diagonal semistability of acyclic matrices. Linear and multilinear algebra 22, 267-283 (1988) · Zbl 0663.15005
[28] Hershkowitz, D.; Berman, A.: Localization of the spectra of P- and P0-matrices. Linear algebra appl. 52/53, 383-397 (1983) · Zbl 0516.15009
[29] Hershkowitz, D.; Berman, A.: Notes on ${\omega}$- and ${\tau}$-matrices. Linear algebra appl. 58, 169-183 (1984) · Zbl 0543.15014
[30] Hershkowitz, D.; Schneider, H.: Scalings of vector spaces and the uniqueness of Lyapunov scaling factors. Linear and multilinear algebra 17, 203-226 (1985) · Zbl 0593.15016
[31] Hershkowitz, D.; Schneider, H.: Lyapunov diagonal semistability of real H-matrices. Linear algebra appl. 71, 119-149 (1985) · Zbl 0577.15015
[32] Hershkowitz, D.; Schneider, H.: Semistability factors and semifactors. Contemp. math. 47, 203-216 (1985) · Zbl 0576.15012
[33] Hershkowitz, D.; Shasha, D.: Cones of real positive semidefinite matrices associated with matrix stability. Linear and multilinear algebra 23, 165-181 (1988) · Zbl 0644.15012
[34] Hurwitz, A.: Über die bedingungen, unter welchen eine gleichung nur wurzeln mit negativen reellen teilen besitzt. Math. ann. 46, 273-284 (1895) · Zbl 26.0119.03
[35] Johnson, C. R.: Second, third and fourth order D-stability. J. res. Nat. bur. Standards sect. B 78, 11-13 (1974) · Zbl 0283.15005
[36] Johnson, C. R.: Sufficient conditions for D-stability. J. econom. Theory 9, 53-62 (1974)
[37] Kaszkurewicz, E.; Hsu, L.: On two classes of matrices with positive diagonal solutions to the Lyapunov equation. Linear algebra appl. 59, 19-27 (1984) · Zbl 0538.15006
[38] Kellogg, R. B.: On complex eigenvalues of M and P matrices. Numer. math. 19, 170-175 (1972) · Zbl 0225.15014
[39] Liénard; Chipart: Sur la signe de la partie réelle des racines d’une équation algébrique. J. math. Pures appl. 10, No. 6, 291-346 (1914) · Zbl 45.1226.03
[40] Lyapunov, A. M.: Le probléme général de la stabilité du mouvement. Ann. math. Stud. 17 (1949)
[41] Mehrmann, V.: On classes of matrices containing M-matrices, totally nonnegative and Hermitian positive semidefinite matrices. Ph.d. dissertation (1982) · Zbl 0508.15009
[42] Metzler, L.: Stability of multiple markets: the Hick conditions. Econometrica 13, 277-292 (1945) · Zbl 0063.03906
[43] Routh, E. J.: A treatise on the stability of a given state of motion. (1877)
[44] Shasha, D.; Berman, A.: On the uniqueness of the Lyapunov scaling factors. Linear algebra appl. 91, 53-63 (1987) · Zbl 0627.15007
[45] Shasha, D.; Berman, A.: More on the uniqueness of the Lyapunov scaling factors. Linear algebra appl. 107, 253-273 (1988) · Zbl 0661.15015
[46] Shasha, D.; Hershkowitz, D.: Maximal Lyapunov scaling factors and their applications in the study of Lyapunov diagonal semistability of block triangular matrices. Linear algebra appl. 103, 21-39 (1988) · Zbl 0676.15009
[47] Taussky, O.: Research problem. Bull. amer. Math. soc. 64, 124 (1958)
[48] Varga, R. S.: Recent results in linear algebra and its applications. Proceedings of the third seminar on methods of numerical applied mathematics, 5-15 (1978) · Zbl 0435.15002