Page 296 - Advanced engineering mathematics

276 CHAPTER 9 Eigenvalues, Diagonalization, and Special Matrices

C k is the circle centered at the kth diagonal element a kk of A, having radius equal to the
sum of the magnitudes of the elements across row k, excluding the diagonal element occurring
in that row.

EXAMPLE 9.7
Let
⎛ ⎞
12i 1 3
A = ⎝ 2 −6 2 + i ⎠ .
3 1 5
The characteristic polynomial of A is

3
p A (λ) =λ + (1 − 12i)λ 2
− (43 + 13i)λ − 68 + 381i.

The Gershgorin circles have centers and radii:

C 1 : (0,12),r 1 = 1 + 3 = 4,
√
C 2 : (−6,0),r 2 = 2 + 5
C 3 : (5,0),r 3 = 3 + 1 = 4.

Figure 9.1 shows these Gershgorin circles. The eigenvalues are in the disks determined by these
circles.

Gershgorin’s theorem is not a way of approximating eigenvalues, since some of the disks
may have large radii. However, sometimes important information that is revealed by these disks
can be useful. For example, in studies of the stability of fluid flow it is important to know whether
eigenvalues occur in the right half-plane.

y

(0, 12)

x
(–6, 0) (5, 0)

FIGURE 9.1 Gerschgorin circles in Example 9.7.

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October 14, 2010 14:49 THM/NEIL Page-276 27410_09_ch09_p267-294