Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 

 

Norms and the localization of roots of matrices


Author: A. S. Householder
Journal: Bull. Amer. Math. Soc. 74 (1968), 816-830
MathSciNet review: 0231526
Full-text PDF

References | Additional Information

References [Enhancements On Off] (What's this?)

  • 1. F. L. Bauer, Theory of norms, Computer Sci. Dept. Stanford Univ. Tech Report No. CS 75, 1967.
  • 2. F. L. Bauer and C. T. Fike, Norms and exclusion theorems, Numer. Math. 2 (1960), 137–141. MR 0118729
  • 3. F. L. Bauer and A. S. Householder, Moments and characteristic roots, Numer. Math. 2 (1960), 42–53. MR 0110188
  • 4. F. L. Bauer and A. S. Householder, Absolute norms and characteristic roots, Numer. Math. 3 (1961), 241–246. MR 0130105
  • 5. J. L. Brenner, New root-location theorems for partitioned matrices, SIAM J. Appl. Math. 16 (1968), 889–898. MR 0239738
  • 6. H. Bückner, Die praktische Behandlung von Integral-Gleichungen, Ergebnisse der angewandten Mathematik. Bd. 1, Springer-Verlag, Berlin, Göttingen, Heidelberg, 1952 (German). MR 0049661
  • 7. V. N. Faddeeva, Computational methods of linear algebra, Dover Publications, Inc., New York, 1959. MR 0100344
  • 8. David G. Feingold and Richard S. Varga, Block diagonally dominant matrices and generalizations of the Gerschgorin circle theorem, Pacific J. Math. 12 (1962), 1241–1250. MR 0151473
  • 9. Miroslav Fiedler, Some estimates of spectra of matrices, Symposium on the numerical treatment of ordinary differential equations, integral and integro-differential equations (Rome, 1960) Birkhäuser, Basel, 1960, pp. 33–36. MR 0134467
  • 10. Miroslav Fiedler and Vlastimil Pták, Some inequalities for the spectrum of a matrix, Mat.-Fyz. Časopis. Slovensk. Akad. Vied. 10 (1960), 148–166 (English, with Russian summary). MR 0124334
  • 11. Miroslav Fiedler and Vlastimil Pták, Generalized norms of matrices and the location of the spectrum, Czechoslovak Math. J. 12 (87) (1962), 558–571 (English, with Russian summary). MR 0146667
  • 12. Noël Gastinel, Matrices du second degré et normes générales en analyse numérique linéaire, Publ. Sci. Tech. Ministère de l’Air, Notes Tech. No. 110, Paris, 1962 (French). MR 0170459
  • 13. S. Gerschgorin, Über die Abgrenzung der Eigenwerte einer Matrix, Izv. Akad. Nauk SSSR. Otd. Mat. Estestv. Nauk, (1931), 749-754.
  • 14. Harold Hotelling, Some new methods in matrix calculation, Ann. Math. Statistics 14 (1943), 1–34. MR 0007851
  • 15. A. S. Householder, On norms of vectors and matrices, Rep. ORNL 1756, Oak Ridge National Laboratory, Oak Ridge, Tenn., 1954. MR 0064008
  • 16. Alston S. Householder, On the convergence of matrix iterations, J. Assoc. Comput. Mach. 3 (1956), 314–324. MR 0082185
  • 17. A. S. Householder, The approximate solution of matrix problems, J. Assoc. Comput. Mach. 5 (1958), 205–243. MR 0128607
  • 18. Alston S. Householder, Minimal matrix norms, Monatsh. Math. 63 (1959), 344–350. MR 0107657
  • 19. A. S. Householder, Localization of the characteristic roots of matrices, Recent Advances in Matrix Theory (Proc. Advanced Seminar, Math. Res. Ce nter, U.S. Army, Univ. Wisconsin, Madison, Wis., 1963) Univ. Wisconsin Press, Madison, Wis., 1964, pp. 39–60. MR 0169856
  • 20. Alston S. Householder, The theory of matrices in numerical analysis, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1964. MR 0175290
  • 21. A. S. Householder, Separation theorems for normalizable matrices, Numer. Math. 9 (1966), 46–50. MR 0212036
  • 22. A. S. Householder, Moments and characteristic roots. II, Numer. Math. 11 (1968), 126–128. MR 0226842
  • 23. W. Kahan, Inclusion theorems for clusters of eigenvalues of Hermitian matrices, Computer Science, Univ. of Torotonto Press, Toronto, 1967.
  • 24. L. Kantorovitch, The method of successive approximations for functional equations, Acta Math. 71 (1939), 63–97. MR 0000095
  • 25. A. Kolmogoroff, Zur Normierbarkeit eines allgemeinen topologischen linearen Raumes, Studia Math. 5 (1934), 29-33.
  • 26. N. J. Lehmann, Optimale Eigenwerteinschliessungen, Numer. Math. 5 (1963), 246–272 (German). MR 0155425
  • 27. Lucien Levy, Sur la possibilité de l'équilibre électrique, C. R. Acad. Sci. Paris 93 (1881), 706-708.
  • 28. Ju. I. Ljubič, Operator norms of matrices, Uspehi Mat. Nauk 18 (1963), no. 4 (112), 161–164 (Russian). MR 0154129
  • 29. A. T. Lonseth, The propagation of error in linear problems, Trans. Amer. Math. Soc. 62 (1947), 193–212. MR 0022315, 10.1090/S0002-9947-1947-0022315-4
  • 30. A. M. Ostrowski, A regularity condition for a class of partitioned matrices, Univ. of Wisconsin MRC Technical Summary Report No. 132, Madison, Wis., 1960.
  • 31. A. M. Ostrowski, On some metrical properties of operator matrices and matrices partitioned into blocks, J. Math. Anal. Appl. 2 (1961), 161–209. MR 0130561
  • 32. A. M. Ostrowski, On some metrical properties of operator matrices and matrices partitioned into blocks, J. Math. Anal. Appl. 2 (1961), 161–209. MR 0130561
  • 33. A. M. Ostrowski, A regularity condition for a class of partitioned matrices, Compositio Math. 15 (1961), 23–27 (1961). MR 0136622
  • 34. F. Robert, Normes vectorielles de vecteurs et de matrices, Rev. Française Traitement Information Chiffres 7 (1964), 261–299 (French). MR 0205433
  • 35. François Robert, Sur les normes vectorielles régulières sur un espace vectoriel de dimension finie, C. R. Acad. Sci. Paris 260 (1965), 5173–5176 (French). MR 0192317
  • 36. F. Robert, Recherche d’une 𝑀-matrice parmi les minorantes d’un opérateur linéaire, Numer. Math. 9 (1966), 189–199 (French). MR 0208821
  • 37. Hans Schneider and W. Gilbert Strang, Comparision theorems for supremum norms, Numer. Math. 4 (1962), 15–20. MR 0132070
  • 38. Josef Stoer, A characterization of Hölder norms, J. Soc. Indust. Appl. Math. 12 (1964), 634–648. MR 0172457
  • 39. Josef Stoer, On the characterization of least upper bound norms in matrix space, Numer. Math. 6 (1964), 302–314. MR 0168571
  • 40. Olga Taussky, A recurring theorem on determinants, Amer. Math. Monthly 56 (1949), 672–676. MR 0032557
  • 41. Helmut Wielandt, On eigenvalues of sums of normal matrices, Pacific J. Math. 5 (1955), 633–638. MR 0073558


Additional Information

DOI: https://doi.org/10.1090/S0002-9904-1968-12051-8