Gershgorin theorem

The famous Gershgorin theorem gives estimates of eigenvalues. The estimates of correspondent eigenvectors are not so well-known. In the chapter we use some estimates of eigenvectors of kinetic matrices. Here we formulate and prove these estimates for general matrices. Below A = (a,y) is a complex n x n matrix. Pi — (sums of nondiagonal elements in rows), Qi — (sums of... 

Gershgorin theorem (Marcus and Mine, 1992, p. 146) The characteristic roots of A lie in the closed region of the z-plane... 

The convergence condition stated above is difQcult to ascertain, since it requires eigenvalues of the Jacobian to be evaluated at the root. A relatively loose condition is to ensure that the eigenvalues of the Jacobian are less than unity in the interval where the expected root is located. Using this condition and the application of the Gershgorin theorem (BeU, 1965), the condition for the convergence of iterations can be ensured for the function g(x). 

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. 

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

Proof. Since A has nonnegative off-diagonal entries and is irreducible, Theorem A.5 asserts that 5(A) is a simple eigenvalue of A, larger than the real parts of all other eigenvalues. The inequality hypothesis and the Gershgorin circle theorem (Theorem A.l) together imply that 5(A) < 0. If 5(A) < 0, then the final assertion of the lemma follows from Theorem A.12. If 5(A) = 0, then Theorem A.5 implies that there exists x > 0 such that Ax = 0. We can assume that Xy < 1 for all j and that x, = 1 for a nonempty subset / of indices. If J is the complementary set of indices then J is non-empty by our assumptions on the row sums of A. For iel we have... 

To answer this question , some mathematical theorems can be useful. In particular, the theorem of localization of the characteristic numbers (or roots) of matrices proved by Gershgorin is most appropriate. This theorem, applied to Hermitian matrices, states that the region of localization of the roots of the secular equation (37) of the order n satisfies the following inequality ("Gershgorin circles" on the complex plane) ... 

The author and cowoikers have introduced a number of theorems from the analytic theory of polynomial equations and perturbation theory for the purpose of gaming irrsight irrto the distribution of eigenvalues by simply knowing the Lanczos parameters. These theorerrrs, which include the Gershgorin circle theorems, enable one to constract spectral domains in the complex plane to which the eigenvalues of L) and are confined. It has been... 

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... 

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