Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(lu) #1

MATRICES AND VECTOR SPACES


Construct an orthonormal set of eigenvectors for the matrix

A=




103


0 − 20


301



.


We first determine the eigenvalues using|A−λI|=0:


0=



∣∣



∣∣


1 −λ 03
0 − 2 −λ 0
301 −λ


∣∣



∣∣=−(1−λ)

(^2) (2 +λ) + 3(3)(2 +λ)
=(4−λ)(λ+2)^2.
Thusλ 1 =4,λ 2 =−2=λ 3. The eigenvectorx^1 =(x 1 x 2 x 3 )Tis found from


103


0 − 20


301






x 1
x 2
x 3


=4




x 1
x 2
x 3


 ⇒ x^1 =√^1
2



1


0


1



.


A general column vector that is orthogonal tox^1 is


x=(ab−a)T, (8.89)

and it is easily shown that


Ax=



103


0 − 20


301






a
b
−a


=− 2




a
b
−a


=− 2 x.

Thusxis a eigenvector ofAwith associated eigenvalue−2. It is clear, however, that there
is an infinite set of eigenvectorsxall possessing the required property; the geometrical
analogue is that there are an infinite number of corresponding vectorsxlying in the
plane that hasx^1 as its normal. We do require that the two remaining eigenvectors are
orthogonal to one another, but this still leaves an infinite number of possibilities. Forx^2 ,
therefore, let us choose a simple form of (8.89), suitably normalised, say,


x^2 =( 010 )T.

The third eigenvector is then specified (to within an arbitrary multiplicative constant)
by the requirement that it must be orthogonal tox^1 andx^2 ; thusx^3 may be found by
evaluating the vector product ofx^1 andx^2 and normalising the result. This gives


x^3 =

1



2


(− 101 )T,


to complete the construction of an orthonormal set of eigenvectors.


8.15 Change of basis and similarity transformations

Throughout this chapter we have considered the vectorxas a geometrical quantity


that is independent of any basis (or coordinate system). If we introduce a basis


ei,i=1, 2 ,...,N, into ourN-dimensional vector space then we may write


x=x 1 e 1 +x 2 e 2 +···+xNeN,
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