Page 295 - Advanced engineering mathematics

9.1 Eigenvalues and Eigenvectors 275

But then
t
(λ − μ)E G = 0.
Since λ = μ, then E G = E · G = 0.
t

EXAMPLE 9.6

⎛ ⎞
3 0 −2
A = ⎝ 0 2 0 ⎠
−20 0
is a 3 × 3 symmetric matrix. The eigenvalues are 2,−1, and 4, with associated eigenvectors

⎛ ⎞ ⎛ ⎞ ⎛ ⎞
0 1 2
, , and .
⎝ 1 ⎠ ⎝ 0 ⎠ ⎝ 0 ⎠
0 2 −1
These eigenvectors are mutually orthogonal.

Finding eigenvalues of a matrix may be difficult because finding the roots of a polynomial
can be difficult. In MAPLE, the command

eigenvals(A);

will list the eigenvalues of A,if n is not too large. The command

eigenvects(A);

will list each eigenvalue, its multiplicity, and, for each eigenvalue, as many linearly inde-
pendent eigenvectors as are associated with that eigenvalue. We can also find the characteristic
polynomial of A by

charpoly(A,t);

in which the variable of the polynomial is called t, but could be given any designation.
There is a method due to Gershgorin that enables us to place the eigenvalues inside disks in
the complex plane. This is sometimes useful to get some idea of how the eigenvalues of a matrix
are distributed.

THEOREM 9.5 Gershgorin

Let A be an n × n matrix of numbers. For k = 1,2,··· ,n let
n

r k = |a kj |.
j=1, j =k
Let C k be the circle of radius r k centered at (α k ,β k ), where a kk = α k + β k i. Then each eigen-
value of A, when plotted as a point in the complex plane, lies on or within one of the circles
C 1 ,··· ,C n .

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

October 14, 2010 14:49 THM/NEIL Page-275 27410_09_ch09_p267-294