The inverse of a totally positive bi-infinite band matrix
Author:
Carl de Boor
Journal:
Trans. Amer. Math. Soc. 274 (1982), 45-58
MSC:
Primary 47B37; Secondary 15A09
DOI:
https://doi.org/10.1090/S0002-9947-1982-0670917-5
MathSciNet review:
670917
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: It is shown that a bounded bi-infinite banded totally positive matrix is boundedly invertible iff there is one and only one bounded sequence mapped by
to the sequence
. The argument shows that such a matrix has a main diagonal, i.e., the inverse of
is the bounded pointwise limit of inverses of finite sections of
principal with respect to a particular diagonal; hence
or its inverse is again totally positive.
- [1] C. de Boor, What is the main diagonal of a biifinite band matrix?, MRC TSR 2049 (1980); in [5,11-23].
- [2] Carl de Boor, Dichotomies for band matrices, SIAM J. Numer. Anal. 17 (1980), no. 6, 894–907. MR 595452, https://doi.org/10.1137/0717074
- [3] Carl de Boor, The inverse of a totally positive bi-infinite band matrix, Trans. Amer. Math. Soc. 274 (1982), no. 1, 45–58. MR 670917, https://doi.org/10.1090/S0002-9947-1982-0670917-5
- [4] Carl de Boor and Allan Pinkus, The approximation of a totally positive band matrix by a strictly banded totally positive one, Linear Algebra Appl. 42 (1982), 81–98. MR 656415, https://doi.org/10.1016/0024-3795(82)90139-2
- [5] Ronald A. DeVore and Karl Scherer (eds.), Quantitative approximation, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 588164
- [6] Samuel Karlin, Total positivity. Vol. I, Stanford University Press, Stanford, Calif, 1968. MR 0230102
- [7] Charles A. Micchelli, Infinite spline interpolation, Approximation in Theorie und Praxis (Proc. Sympos., Siegen, 1979) Bibliographisches Inst., Mannheim, 1979, pp. 209–238. MR 567661
- [8] A. S. Cavaretta Jr., W. Dahmen, C. A. Micchelli, and P. W. Smith, On the solvability of certain systems of linear difference equations, SIAM J. Math. Anal. 12 (1981), no. 6, 833–841. MR 635236, https://doi.org/10.1137/0512069
Retrieve articles in Transactions of the American Mathematical Society with MSC: 47B37, 15A09
Retrieve articles in all journals with MSC: 47B37, 15A09
Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1982-0670917-5
Keywords:
Bi-infinite,
matrix,
total positivity,
inverse,
banded,
main diagonal
Article copyright:
© Copyright 1982
American Mathematical Society