Abstract
In this chapter, we shall first review, in Sect. 3.2, certain results on infinite linear systems whose “coefficient” matrices are diagonally dominant in some sense. We treat these infinite matrices in their own right and also as operators over certain normed linear spaces. These results show the extent to which results that are known for finite matrices have been generalized. Next, in Sect. 3.3, we recall some results on eigenvalues for operators mainly of the type considered in the second section. We also review a powerful numerical method for computing eigenvalues of certain diagonally dominant tridiagonal operators. Section 3.4 concerns linear differential systems whose coefficient matrices are operators on either ℓ 1 or \(\ell^{\infty }\). Convergence results for truncated systems are presented. The concluding section, viz., Sect. 3.5 discusses an iterative method for numerically solving a linear equation whose matrix is treated as an operator on \(\ell^{\infty }\) satisfying certain conditions, including a diagonal dominance condition.
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References
Arley, N., Borchsenius, V.: On the theory of infinite systems of differential equations and their application to the theory of stochastic processes and the perturbation theory of quantum mechanics. Acta Math. 76, 261–322 (1945)
Bellman, R.: The boundedness of solutions of infinite systems of linear differential equations. Duke Math. J. 14, 695–706 (1947)
Farid, F.O.: Spectral properties of perturbed linear operators and their application to infinite matrices. Proc. Am. Math. Soc. 112, 1013–1022 (1991)
Farid, F.O., Lancaster, P.: Spectral properties of diagonally dominant infinite matrices. I. Proc. R. Soc. Edinb. Sect. A 111, 304–314 (1989)
Farid, F.O., Lancaster, P.: Spectral properties of diagonally dominant infinite matrices. II. Linear Algebra Appl. 143, 7–17 (1991)
Kantorovich, L.V., Krylov, V.I.: Approximate Methods of Higher Analysis. Interscience, New York (1964)
Malejki, M.: Approximation and asymptotics of eigenvalues of unbounded self-adjoint Jacobi matrices acting in ℓ 2 by the use of finite submatrices. Cent. Eur. J. Math. 8, 114–128 (2010)
Shaw, L.: Solutions for infinite-matrix differential equations. J. Math. Anal. Appl. 41, 373–383 (1973)
Shivakumar, P.N.: Diagonally dominant infinite matrices in linear equations. Util. Math. 1, 235–248 (1972)
Shivakumar, P.N., Chew, K.H.: Iterations for diagonally dominant matrices. Can. Math. Bull. 19, 375–377 (1976)
Shivakumar, P.N., Williams, J.J.: An iterative method with truncation for infinite linear systems. J. Comp. Appl. Math. 24, 199–207 (1988)
Shivakumar, P.N., Wong, R.: Linear equations in infinite matrices. Linear Algebra Appl. 7, 553–562 (1973)
Shivakumar, P.N., Chew, K.H., Williams, J.J.: Error bounds for the truncation of infinite linear differential systems. J. Inst. Math. Appl. 25, 37–51 (1980)
Shivakumar, P.N., Williams, J.J., Rudraiah, N.: Eigenvalues for infinite matrices. Linear Algebra Appl. 96, 35–63 (1987)
Williams, J.J., Ye, Q.: Infinite matrices bounded on weighted ℓ 1 spaces. Linear Algebra Appl. 438, 4689–4700 (2013)
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Shivakumar, P.N., Sivakumar, K.C., Zhang, Y. (2016). Infinite Linear Equations. In: Infinite Matrices and Their Recent Applications. Springer, Cham. https://doi.org/10.1007/978-3-319-30180-8_3
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DOI: https://doi.org/10.1007/978-3-319-30180-8_3
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