Gershgorin’s theorem

This is Gershgorin s theorem. Moreover, if any A is separated from the others, and the p s small enough so that the particular disk (2-14) is disjoint from all the others, then that disk contains exactly one root. By applying the theorem to the matrix D(A + B)D l, for a suitable matrix D, it is sometimes possible to obtain even better bounds. 

For matrices that are not triangular, we cannot determine the eigenvalues by inspection, but we can obtain upper and lower bounds using Gershgorin s theorem. Let be an iVx W... 

A proof, relying upon the concepts of matrix norm and spectral radius introduced in the next section, is provided in the supplemental material in the accompanying website. Figure 3.3 demonstrates the application of Gershgorin s theorem to a 3 x 3 matrix. test-Gershgorin.m generates this plot for any input matrix. 

Applying Gershgorin s theorem to study the convergence of iterative linear solvers... 

As a demonstration of the useMness of Gershgorin s theorem, we generate a convergence criterion for the Jacobi iterative method of solving Ax = b. This example is typical of the use of eigenvalues in numerical analysis, and also shows why the questions of eigenvector basis set existence raised in the next section are of such importance. 

A.I. From Gershgorin s theorem, derive lower and upper bounds on the possible eigenvalues of the matrix... 

Figure 2.7 The Gershgorin s circle theorem. The three eigenvalues of the matrix A are located within the Gershgorin s circles.

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