Gershgorin circles

Properties (1) and (2) also result from the estimates of the eigenvalues using Gershgorin circles [29] any eigenvalue X of the matrix K lie on a complex plane in one of the circles of type X — ku ku or otherwise... 

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... 

Figure 3.2 The complex plane, showing that an eigenvalue must lie within one of the Gershgorin circles.

The same argument applies to AT and may be used to calculate Gershgorin s circles with respect to rows instead of columns. 

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

The resulting circles are sketched in Fig. 16.5 for the n plot. The circles are called Gershgorin rings. The bands that the circles sweep out are Gershgorin bands. If all the off-diagonal elements were zero, the circles would have zero radius and no interaction would be present. Therefore the bigger the circles, the more interaction is present in the system. 

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