The Gerschgorin discs under unitary similarity
Zalewska-Mitura, Anna ; Zemánek, Jaroslav
Banach Center Publications, Tome 38 (1997), p. 427-441 / Harvested from The Polish Digital Mathematics Library

The intersection of the Gerschgorin regions over the unitary similarity orbit of a given matrix is studied. It reduces to the spectrum in some cases: for instance, if the matrix satisfies a quadratic equation, and also for matrices having "large" singular values or diagonal entries. This leads to a number of open questions.

Publié le : 1997-01-01
EUDML-ID : urn:eudml:doc:208645
@article{bwmeta1.element.bwnjournal-article-bcpv38i1p427bwm,
     author = {Zalewska-Mitura, Anna and Zem\'anek, Jaroslav},
     title = {The Gerschgorin discs under unitary similarity},
     journal = {Banach Center Publications},
     volume = {38},
     year = {1997},
     pages = {427-441},
     zbl = {0877.15020},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv38i1p427bwm}
}
Zalewska-Mitura, Anna; Zemánek, Jaroslav. The Gerschgorin discs under unitary similarity. Banach Center Publications, Tome 38 (1997) pp. 427-441. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv38i1p427bwm/

[000] G. R. Allan and J. Zemánek [to appear], Invariant subspaces for pairs of projections, J. London Math. Soc. | Zbl 0939.47003

[001] H. Auerbach [1933], Sur le nombre de générateurs d'un groupe linéaire borné, C. R. Acad. Sci. Paris 197, 1385-1386. | Zbl 59.0434.02

[002] B. Aupetit [1991], A Primer on Spectral Theory, Springer, New York.

[003] B. Aupetit, E. Makai, Jr. and J. Zemánek [1996], Strict convexity of the singular value sequences, Acta Sci. Math. (Szeged) 62, 517-521. | Zbl 0880.47012

[004] F. L. Bauer and C. T. Fike [1960], Norms and exclusion theorems, Numer. Math. 2, 137-141. | Zbl 0101.25503

[005] H. E. Bell [1965], Gershgorin's theorem and the zeros of polynomials, Amer. Math. Monthly 72, 292-295. | Zbl 0134.02003

[006] E. Bodewig [1956], Matrix Calculus, North-Holland, Amsterdam. | Zbl 0086.32501

[007] E. T. Browne [1928], The characteristic equation of a matrix, Bull. Amer. Math. Soc. 34, 363-368. | Zbl 54.0109.04

[008] E. T. Browne [1939], Limits to the characteristic roots of a matrix, Amer. Math. Monthly 46, 252-265. | Zbl 0021.09904

[009] E. T. Browne [1958], Introduction to the Theory of Determinants and Matrices, University of North Carolina Press, Chapel Hill, NC. | Zbl 0079.01201

[010] R. A. Brualdi and S. Mellendorf [1994], Regions in the complex plane containing the eigenvalues of a matrix, Amer. Math. Monthly 101, 975-985. | Zbl 0838.15010

[011] R. L. Causey [1958], Computing eigenvalues of non-Hermitian matrices by methods of Jacobi type, J. Soc. Indust. Appl. Math. 6, 172-181. | Zbl 0097.32701

[012] G. Dahlquist [1958], Stability and Error Bounds in the Numerical Integration of Ordinary Differential Equations, dissertation, Uppsala. Published in Kungl. Tekn. Högsk. Hand. Stockholm, No. 130, 1959.

[013] C. Davis [1955], Generators of the ring of bounded operators, Proc.Amer.Math.Soc.6, 970-972. | Zbl 0066.09801

[014] J.Dazord [1991], Sur une norme de matrices, C.R.Acad.Sci.ParisSér. I Math. 312, 597-600. | Zbl 0724.47006

[015] J. Dazord [1994], On the C-numerical range of a matrix, Linear Algebra Appl. 212/213, 21-29. | Zbl 0814.15020

[016] J. Dazord [1995a], Une propriété extremale de la diagonale d'une matrice, lecture notes, Luminy.

[017] J. Dazord [1995b], Matrices (1-d), lecture notes, Luminy.

[018] J. Dazord [1996], Trace norm and spatial radius of a matrix, lecture notes, Chemnitz.

[019] R. Gabriel [1979], Matrizen mit maximaler Diagonale bei unitärer Similarität, J. Reine Angew. Math. 307/308, 31-52. | Zbl 0396.15010

[020] N. Gastinel [1960], Utilisation de matrices vérifiant une équation de degré 2 pour la transmutation de matrices, C. R. Acad. Sci. Paris 250, 1960-1961. | Zbl 0091.12002

[021] S. Gerschgorin [1931], Über die Abgrenzung der Eigenwerte einer Matrix, Izv. Akad. Nauk SSSR 7, 749-754. | Zbl 0003.00102

[022] Y. Gu [1994], The distribution of eigenvalues of a matrix, Acta Math. Appl. Sinica 17, 501-511 (in Chinese).

[023] K. E. Gustafson and D. K. M. Rao [1997], Numerical Range, Springer, New York.

[024] P. R. Halmos [1995], Linear Algebra Problem Book, Mathematical Association of America, Washington, DC.

[025] T. Hawkins [1975], Cauchy and the spectral theory of matrices, Historia Math. 2, 1-29. | Zbl 0296.01014

[026] R. A. Horn and C. R. Johnson [1985], Matrix Analysis, Cambridge University Press, Cambridge. | Zbl 0576.15001

[027] R. A. Horn and C. R. Johnson [1991], Topics in Matrix Analysis, Cambridge University Press, Cambridge. | Zbl 0729.15001

[028] A. S. Householder [1964], The Theory of Matrices in Numerical Analysis, Blaisdell, New York. | Zbl 0161.12101

[029] T. J. Laffey [1981], Algebras generated by two idempotents, Linear Algebra Appl. 37, 45-53. | Zbl 0459.16010

[030] P. Lascaux et R. Théodor [1993], Analyse Numérique Matricielle Appliquée à l'Art de l'Ingénieur 1, Masson, Paris. | Zbl 0601.65017

[031] L. László [1991], Upper bounds for matrix diagonals, Linear and Multilinear Algebra 30, 283-301. | Zbl 0745.15011

[032] L. László [1996], Upper bounds for the best normal approximation, lecture notes, Chemnitz.

[033] L. László [1997], Upper bounds for the best normal approximation, preprint.

[034] S. L. Lee [1996], Best available bounds for departure from normality, SIAM J. Matrix Anal. Appl. 17, 984-991. | Zbl 0877.65024

[035] S. M. Lozinskiĭ [1958], Error estimate for numerical integration of ordinary differential equations I, Izv. Vyssh. Uchebn. Zaved. Mat., no. 5 (6), 52-90; errata, 1959, no. 5 (12), 222 (in Russian). | Zbl 0198.21202

[036] E. H. Luchnis and M. A. McLoughlin [1996], In memoriam: Olga Taussky-Todd, Notices Amer. Math. Soc. 43, 838-847. | Zbl 1044.01541

[037] G. Lumer [1961], Semi-inner-product spaces, Trans. Amer. Math. Soc. 100, 29-43. | Zbl 0102.32701

[038] C. C. MacDuffee [1946], The Theory of Matrices, Chelsea, New York. | Zbl 0007.19507

[039] M. Marcus and H. Minc [1964], A Survey of Matrix Theory and Matrix Inequalities, Allyn and Bacon, Boston. | Zbl 0126.02404

[040] M. Marcus and M. Sandy [1985], Singular values and numerical radii, Linear and Multilinear Algebra 18, 337-353. | Zbl 0592.15009

[041] M. Marden [1966], Geometry of Polynomials, American Mathematical Society, Providence, RI. | Zbl 0162.37101

[042] L. Mirsky [1955], An Introduction to Linear Algebra, Clarendon Press, Oxford. | Zbl 0066.26305

[043] M. Newman [1980]. Geršgorin revisited, Linear Algebra Appl. 30, 247-249. | Zbl 0438.15011

[044] N. Nirschl and H. Schneider [1964], The Bauer fields of values of a matrix, Numer. Math. 6, 355-365. | Zbl 0126.32102

[045] N. Obreškov [1963], Zeros of Polynomials, Izdat. Bŭlgar. Akad. Nauk, Sofia (in Bulgarian). | Zbl 1248.33001

[046] W. V. Parker [1948], Sets of complex numbers associated with a matrix, Duke Math. J. 15, 711-715. | Zbl 0031.14803

[047] W. V. Parker [1951], Characteristic roots and field of values of a matrix, Bull. Amer. Math. Soc. 57, 103-108. | Zbl 0042.25101

[048] M. Parodi [1959], La Localisation des Valeurs Caractéristiques des Matrices et Ses Applications, Gauthier-Villars, Paris. | Zbl 0087.01602

[049] S. Prasanna [1981], The norm of a derivation and the Björck-Thomee-Istratescu theorem, Math. Japon. 26, 585-588. | Zbl 0475.47007

[050] V. V. Prasolov [1994], Problems and Theorems in Linear Algebra, American Mathematical Society, Providence, RI. | Zbl 0803.15001

[051] V. Pták et J. Zemánek [1976], Continuité lipschitzienne du spectre comme fonction d'un opérateur normal, Comment. Math. Univ. Carolin. 17, 507-512. | Zbl 0341.47019

[052] H. Radjavi and P. Rosenthal [1970], Matrices for operators and generators of B(H), J. London Math. Soc. (2) 2, 557-560. | Zbl 0197.10801

[053] A. G. Robertson [1974], A note on the unit ball in C*-algebras, Bull. London Math. Soc. 6, 333-335. | Zbl 0291.46042

[054] V. Scharnitzky [1996], Matrix Calculus, Műszaki Könyvkiadó, Budapest (in Hungarian).

[055] I. Schur [1909], Über die charakteristischen Wurzeln einer linearen Substitution mit einer Anwendung auf die Theorie der Integralgleichungen, Math. Ann. 66, 488-510. | Zbl 40.0396.03

[056] H. Shapiro [1991], A survey of canonical forms and invariants for unitary similarity, Linear Algebra Appl. 147, 101-167. | Zbl 0723.15007

[057] K. Skurzyński [1996], Elements of the theory of matrices, Gradient, no. 4, 216-234 (in Polish).

[058] A. Smoktunowicz [1996], Remarks on inclusion theorems for normal matrices, lecture notes, Warszawa.

[059] J. G. Stampfli and J. P. Williams [1968], Growth conditions and the numerical range in a Banach algebra, Tôhoku Math. J. 20, 417-424. | Zbl 0175.43902

[060] P. Stein [1952], A note on bounds of multiple characteristic roots of a matrix, J. Research Nat. Bur. Standards 48, 59-60. | Zbl 0049.01002

[061] E. L. Stolov [1979], The Hausdorff set of a matrix, Izv. Vyssh. Uchebn. Zaved. Mat., no. 10, 98-100 (in Russian). | Zbl 0428.15011

[062] B.-S. Tam [1986], A simple proof of the Goldberg-Straus theorem on numerical radii, Glasgow Math. J. 28, 139-141. | Zbl 0605.15019

[063] O. Taussky [1948], Bounds for characteristic roots of matrices, Duke Math. J. 15, 1043-1044. | Zbl 0031.24405

[064] O. Taussky [1949], A recurring theorem on determinants, Amer. Math. Monthly 56, 672-676. | Zbl 0036.01301

[065] O. Taussky [1962], Eigenvalues of finite matrices: Some topics concerning bounds for eigenvalues of finite matrices, in: Survey of Numerical Analysis (ed. J. Todd), McGraw-Hill, New York, 279-297.

[066] R. C. Thompson [1987], The matrix numerical range, Linear and Multilinear Algebra 21, 321-323. | Zbl 0637.15019

[067] N.-K. Tsing [1983], Diameter and minimal width of the numerical range, Linear and Multilinear Algebra 14, 179-185. | Zbl 0533.15017

[068] H. W. Turnbull and A. C. Aitken [1932], An Introduction to the Theory of Canonical Matrices, Blackie, London. | Zbl 0005.19303

[069] R. S. Varga [1962], Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ.

[070] R. S. Varga [1965], Minimal Gerschgorin sets, Pacific J. Math. 15, 719-729. | Zbl 0168.02904

[071] T. Yoshino [1993], Introduction to Operator Theory, Longman, Harlow.

[072] A. Zalewska-Mitura [1997], Localization of the Spectrum of Matrices by Means of Unitary Similarities, dissertation, Institute of Mathematics of the Polish Academy of Sciences, Warszawa (in Polish).