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Infinite Linear Equations

In: Infinite Matrices and Their Recent Applications

Author

Listed:
  • P. N. Shivakumar

    (University of Manitoba, Department of Mathematics)

  • K. C. Sivakumar

    (Indian Institute of Technology, Madras, Department of Mathematics)

  • Yang Zhang

    (University of Manitoba, Department of Mathematics)

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 ℓ ∞ $$\ell^{\infty }$$ . 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 ℓ ∞ $$\ell^{\infty }$$ satisfying certain conditions, including a diagonal dominance condition.

Suggested Citation

  • P. N. Shivakumar & K. C. Sivakumar & Yang Zhang, 2016. "Infinite Linear Equations," Springer Books, in: Infinite Matrices and Their Recent Applications, chapter 0, pages 27-39, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-30180-8_3
    DOI: 10.1007/978-3-319-30180-8_3
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