Structural and computational properties of possibly singular semiseparable matrices

A classical result of structured numerical linear algebra states that the inverse of a nonsingular semiseparable matrix is a tridiagonal matrix. Such a property of a semiseparable matrix has been proved to be useful for devising linear complexity solvers, for establishing recurrence relations among...

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Bibliographic Details
Published in:Linear algebra and its applications Vol. 340; no. 1-3; pp. 183 - 198
Main Authors: Fasino, D., Gemignani, L.
Format: Journal Article
Language:English
Published: New York, NY Elsevier Inc 01.01.2002
Elsevier Science
Subjects:
Algebra
Block tridiagonal matrices
Exact sciences and technology
Linear and multilinear algebra, matrix theory
Linear and multilinear algebra, matrix theory (finite and infinite)
Linear complexity algorithms
Logic, set theory, and algebra
Mathematical methods in physics
Mathematics
Physics
Sciences and techniques of general use
Semiseparable matrices
Linear complexity algorithms
Block tridiagonal matrices
Semiseparable matrices
Singular matrix
Singularity
Recurrence relation
Linear algebra
Characteristic polynomial
Sparse matrix
Numerical method
Rank
Tridiagonal matrix
Generalized inverse
ISSN:0024-3795, 1873-1856
Online Access:Get full text
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Summary:A classical result of structured numerical linear algebra states that the inverse of a nonsingular semiseparable matrix is a tridiagonal matrix. Such a property of a semiseparable matrix has been proved to be useful for devising linear complexity solvers, for establishing recurrence relations among its columns or rows and, moreover, for efficiently evaluating its characteristic polynomial. In this paper, we provide sparse structured representations of a semiseparable matrix A which hold independently of the fact that A is singular or not. These relations are found by pointing out the band structure of the inverse of the sum of A plus a certain sparse perturbation of minimal rank. Further, they can be used to determine in a computationally efficient way both a reflexive generalized inverse of A and its characteristic polynomial.
ISSN:0024-3795
1873-1856
DOI:10.1016/S0024-3795(01)00404-9