Abstract
In this expository paper we illustrate the role that the field of values (or numerical range) of a matrix plays in connection with certain problems of numerical analysis. These include the approximation of matrix functions and the convergence of preconditioned Krylov subspace methods for solving large systems of equations arising from the discretization of partial differential equations.
Highlights
The field of values, or numerical range, of a matrix is a well-studied object in linear algebra and functional analysis [28]
The field of values has become increasingly important in numerical analysis, in particular in certain problems of numerical linear algebra involving functions of matrices and iterative methods for solving large systems of linear equations
Analyzing the behavior of algorithms for the approximation of functions of such matrices as their size increases is of central importance in numerical linear algebra
Summary
The field of values, or numerical range, of a matrix (or operator in Hilbert space) is a well-studied object in linear algebra and functional analysis [28]. The field of values has become increasingly important in numerical analysis, in particular in certain problems of numerical linear algebra involving functions of matrices and iterative methods for solving large systems of linear equations In such problems one has to deal with sequences of matrices of increasing (potentially unbounded) dimension. Where κ(X ) = inf X X −1 , where the infimum is taken over all nonsingular matrices X that diagonalize A; note that κ(X ) ≥ 1, and that κ(X ) = 1 when A is normal This quantity is known as the spectral condition number of the eigenbasis of A. The eigenvalues give at best a partial picture of the underlying behavior Another limitation of spectral analysis is that the eigenvalues of a non-normal matrix can be highly sensitive to perturbations. In the rest of the paper we will focus on two such problems from the field of numerical linear algebra
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