Page 293 - Advanced engineering mathematics

9.1 Eigenvalues and Eigenvectors 273

But then c 1 V 1 = O. Since an eigenvalue cannot be the zero vector, this means that c 1 = 0also.
Therefore V 1 ,...,V k are linearly independent. By induction, this proves the theorem.

In Example 9.3, the 3 × 3matrix A had only two distinct eigenvalues, and only two lin-
early independent eigenvectors. However, the matrix of the next example has three linearly
independent eigenvectors even though it has only two distinct eigenvalues. When eigenvalues
are repeated, a matrix may or may not have n linearly independent eigenvectors.

EXAMPLE 9.5
Let
⎛ ⎞
5 −4 4
A = 12 −11 12 ⎠ .
⎝
4 −4 5
The eigenvalues of A are −3,1,1, with 1 a repeated root of the characteristic polynomial.
Corresponding to −3, we find an eigenvector
⎛ ⎞
1
.
⎝ 3 ⎠
1
Now look for an eigenvector corresponding to 1. We must solve the system
⎛ ⎞⎛ ⎞ ⎛ ⎞
−4 4 −4 x 1 0
((1)I 2 − A)X = −12 12 −12 ⎠⎝ x 2 ⎠ = 0 ⎠ .
⎝
⎝
−4 4 −4 x 3 0
This system has the general solution
1 0
⎛ ⎞ ⎛ ⎞
α ⎝ 0 ⎠ + β ⎝ 1 ⎠ ,
−1 1
in which α and β are any numbers. With α = 1 and β = 0, and then with α = 0 and β = 1, we
obtain two linearly independent eigenvectors associated with eigenvalue 1:
⎛ ⎞ ⎛ ⎞
1 0
and .
⎝ 0 ⎠ ⎝ 1 ⎠
−1 1
For this matrix A, we can produce three linearly independent eigenvectors, even though the
eigenvalues are not distinct.

Eigenvalues and eigenvectors of special classes of matrices may exhibit special properties.
Symmetric matrices form one such class. A =[a ij ] is symmetric if a ij = a ji whenever i = j.This
t
means that A=A , hence that each off-diagonal element is equal to its reflection across this main
diagonal. For example,
⎛ ⎞
−7 −2 − i 1 14
⎜ −2 − i 2 −947i ⎟
⎜ ⎟
1 −9 −4
⎝ π ⎠
14 47i π 22
is symmetric.
It is a significant property of symmetric matrices that those with real elements have all real
eigenvalues.

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October 14, 2010 14:49 THM/NEIL Page-273 27410_09_ch09_p267-294