
Accession Number : AD0688658
Title : BOUNDS AND LOCATION THEOREMS OF THE EIGENVALUES OF MATRICES PARTITIONED INTO BLOCKS.
Descriptive Note : Interim technical rept. no. 21,
Corporate Author : TEXAS UNIV AUSTIN COMPUTATION CENTER
Personal Author(s) : Kim,Gil Chang
Report Date : MAY 1969
Pagination or Media Count : 83
Abstract : The wellknown Gerschgorin circle theorem and the subsequent result that the maximum row sum, which is equal to the maximum norm, of an arbitrary square matrix A is an upper bound of the spectral radius depend on the absolute values of the elements of A. Essentially, these results arise from theorems which establish the nonsingularity of A. For example, it is known that A is nonsingular if A is strictly diagonally dominant and the application of this result to A  zI yields the Gerschgorin circle theorem. To generalize this, A is considered to be a matrix partitioned into blocks, where it is assumed that the diagonal blocks are square and nonsingular. Then A is nonsingular if the matrix (A) is an Mmatrix, which is formed by replacing the diagonal blocks in A by their infimums and the offdiagonal blocks by their norms. In a similar manner, the previous bound of the spectral radius can be generalized. These new results can give significant improvements over the usual Gerschgorin circles in providing bounds for the eigenvalues of A under a suitable choice of norms and partitioning. Restricting the choice of norms to the sum norm, the Euclidean norm, and the maximum norm, it is found that improvements are always possible for a class of matrices arising from the numerical solution of partial differential equations. (Author)
Descriptors : (*MATRICES(MATHEMATICS), THEOREMS), PARTIAL DIFFERENTIAL EQUATIONS, NUMERICAL ANALYSIS, THEOREMS, THESES
Subject Categories : Theoretical Mathematics
Distribution Statement : APPROVED FOR PUBLIC RELEASE