BOUNDS AND LOCATION THEOREMS OF THE EIGENVALUES OF MATRICES PARTITIONED INTO BLOCKS.
Interim technical rept. no. 21,
TEXAS UNIV AUSTIN COMPUTATION CENTER
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The well-known 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 non-singular. Then A is non-singular if the matrix A is an M-matrix, which is formed by replacing the diagonal blocks in A by their infimums and the off-diagonal 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
- Theoretical Mathematics