Eigenvalue localization refinements for the Schur complement
Introduction
The starting point for the considerations that follow is the fact that a nonsingular matrix A is an H-matrix if and only if there exists a diagonal nonsingular matrix W such that AW is a strictly diagonally dominant (SDD) matrix. In other words, see [5], the class of H-matrices is diagonally derived from the class of SDD matrices. Some special subclasses of H-matrices could be characterized by the form of the corresponding scaling matrix W. These characterizations will be recalled in the first section, for they will be used to obtain some new results on the eigenvalue distribution of the Schur complement.
Throughout the paper we will use the following notations:Obviously, for arbitrary subset S and each index ,
It is important to emphasize that all the time we are dealing with nonsingular H-matrices, calling them shortly H-matrices. To be precise, we recall the definition of SDD matrices and H-matrices, as well as some more preliminaries.
Section snippets
Preliminaries
Definition 1.1 A matrix is called an SDD matrix if, for each , it holds that Theorem 1.2 If a matrix is an SDD matrix, then it is nonsingular, moreover it is an H-matrix.
That SDD matrices are nonsingular is an old and recurring result in matrix theory. In [12] it is pointed out that this nonsingularity result and the well-known Geršgorin’s Theorem (1931) that gives the eigenvalue localization are actually equivalent. Theorem 1.3 Geršgorin Theorem For any matrix and any , there is a positive integer k in N
Vertical eigenvalue bands
The Schur complement of A with respect to a proper subset of , is denoted by and defined to bewhere stands for the submatrix of lying in the rows indexed by α and the columns indexed by , while is abbreviated to . Troughout the paper we assume that is a nonsingular matrix.
As stated in [8], investigating the distribution for the eigenvalues of the Schur complement is of great significance. If the eigenvalues of the Schur
Numerical examples
Example 1.
The given matrix, A, is for . We take , and determine the ‘good’ interval for γ. It is easy to see that belongs to this ‘good’ interval. For the chosen γ, we scale the matrix A, and, as in Theorem 2.2, obtain the vertical band () given in the Fig. 1.
Example 2.
Matrix B is for . For , we determine the
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