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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Some properties of a class of band matrices


Authors: W. D. Hoskins and P. J. Ponzo
Journal: Math. Comp. 26 (1972), 393-400
MSC: Primary 65F30
MathSciNet review: 0303703
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ A(2r + 1,n)$ denote the $ n \times n$ band matrix, of bandwidth $ 2r + 1$, with the binomial coefficients in the expansion of $ {(x - 1)^{2r}}$ as the elements in each row and column. Using the fact that the rows of $ A(2r + 1,n)$ provide the coefficients for the $ 2r$th central difference, a number of properties of $ A(2r + 1,n)$ are obtained for all positive integers $ r$ and $ n$. These include obtaining explicit formulas for $ \det A(2r + 1,n),{A^{ - 1}}(2r + 1,n),\vert\vert{A^{ - 1}}(2r + 1,n)\vert{\vert _\infty }$ and an upper triangular matrix $ U$ such that $ A(2r + 1,n)U$ is lower triangular.


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

  • [1] R. G. Stanton & D. A. Sprott, ``Some finite inversion formulae,'' Math. Gazette, v. 46, 1962, pp. 197-202.
  • [2] Richard S. Varga, Matrix iterative analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0158502 (28 #1725)
  • [3] Numerical solution of ordinary and partial differential equations., Based on a Summer School held in Oxford, August-September 1961, Pergamon Press, Oxford-London-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass.-Palo Alto, Calif.-London, 1962. MR 0146969 (26 #4488)
  • [4] D. E. Rutherford, Some continuant determinants arising in physics and chemistry. II, Proc. Roy. Soc. Edinburgh. Sect. A. 63 (1952), 232–241. MR 0059232 (15,495d)
  • [5] John Riordan, Combinatorial identities, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0231725 (38 #53)
  • [6] A. C. Aitken, Determinants and Matrices, Oliver & Boyd, London, 1936, p. 124.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1972-0303703-3
PII: S 0025-5718(1972)0303703-3
Keywords: Determinants, band matrices, inverse matrices, norms on inverses
Article copyright: © Copyright 1972 American Mathematical Society