Mathematical Methods for Physics and Engineering : A Comprehensive Guide

8.13 EIGENVECTORS AND EIGENVALUES

writeA†xi=λ∗ixi. From (8.74) and (8.75) we have

(λi−λj)(xi)†xj= 0. (8.76)

Thus,ifλi=λjthe eigenvectorsxiandxjmust be orthogonal,i.e.(xi)†xj= 0.

It follows immediately from (8.76) that if allNeigenvalues of a normal matrix

Aare distinct then allNeigenvectors ofAare mutually orthogonal. If, however,

two or more eigenvalues are the same then further consideration is required. An

eigenvalue corresponding to two or more different eigenvectors (i.e. they are not

simply multiples of one another) is said to bedegenerate. Suppose thatλ 1 isk-fold

degenerate, i.e.

Axi=λ 1 xi fori=1, 2 ,...,k, (8.77)

but that it is different from any ofλk+1,λk+2, etc. Then any linear combination

of thesexiis also an eigenvector with eigenvalueλ 1 ,since,forz=

∑k

i=1cix

i,

Az≡A

∑k

i=1

cixi=

∑k

i=1

ciAxi=

∑k

i=1

ciλ 1 xi=λ 1 z. (8.78)

If thexidefined in (8.77) are not already mutually orthogonal then we can

construct new eigenvectorszithat are orthogonal by the following procedure:

z^1 =x^1 ,

z^2 =x^2 −

[

(zˆ

1

)†x^2

]

zˆ^1 ,

z^3 =x^3 −

[

(zˆ

2

)†x^3

]

zˆ^2 −

[

(ˆz

1

)†x^3

]

zˆ^1 ,

..

.

zk=xk−

[

(zˆk−^1 )†xk

]

zˆk−^1 −···−

[

(zˆ^1 )†xk

]

ˆz^1.

In this procedure, known asGram–Schmidt orthogonalisation, each new eigen-

vectorziis normalised to give the unit vectorzˆibefore proceeding to the construc-

tion of the next one (the normalisation is carried out by dividing each element of

the vectorziby [(zi)†zi]^1 /^2 ). Note that each factor in brackets (ˆzm)†xnis a scalar

product and thus only a number. It follows that, as shown in (8.78), each vector

ziso constructed is an eigenvector ofAwith eigenvalueλ 1 and will remain so

on normalisation. It is straightforward to check that, provided the previous new

eigenvectors have been normalised as prescribed, eachziis orthogonal to all its

predecessors. (In practice, however, the method is laborious and the example in

subsection 8.14.1 gives a less rigorous but considerably quicker way.)

Therefore, even ifAhas some degenerate eigenvalues we canby construction

obtain a set ofNmutually orthogonal eigenvectors. Moreover, it may be shown

(although the proof is beyond the scope of this book) that these eigenvectors

arecompletein that they form a basis for theN-dimensional vector space. As