Anon

Fundamentals of Matrix Algebra 397

To determine the eigenvectors of a matrix and the relative eigenvalues,
consider that the equation Ax=λx can be written as

()AI−λ x=^0

which can, in turn, be written as a system of linear equations:

()−λ =

−λ ⋅⋅

⋅ ⋅⋅⋅⋅

⋅−λ⋅

⋅ ⋅⋅⋅⋅

⋅⋅−λ





      





      

⋅

⋅





     





     

=

aaa

aa a

aaa

x

x

x

AIx 0

jn

iiiin

nnjnn

i

n

1,11,1,

,1 ,,

,1 ,,

1

This system of equations has nontrivial solutions only if the matrix
AI−λ is singular. To determine the eigenvectors and the eigenvalues of the
matrix A we must therefore solve the following equation:

−λ =

−λ ⋅⋅

⋅ ⋅⋅⋅⋅

⋅−λ⋅

⋅ ⋅⋅⋅⋅

⋅⋅−λ

=

aaa

aa a

aaa

AI 0

jn

iiiin

nnjnn

1,11,1,

,1 ,,

,1 ,,

The expansion of this determinant yields a polynomial φλ() of degree
n known as the characteristic polynomial of the matrix A. The equation

φλ()= (^0) is known as the characteristic equation of the matrix A. In general,
this equation will have n roots λs which are the eigenvalues of the matrix A.
To each of these eigenvalues corresponds a solution of the system of linear
equations, illustrated as follows:
−λ ⋅⋅
⋅⋅⋅⋅⋅
⋅−λ⋅
⋅⋅⋅⋅⋅
⋅⋅−λ





      





      

⋅

⋅





      





      

=

aaa

aa a

aaa

x

x

x

0

sj n

iiis in

nnjnns

i

n

1,11,1,

,1 ,,

,1 ,,

(^1) s
s
s
Each solution represents the eigenvector xs corresponding to the eigenvalue
λs. As explained in Chapter 12, the determination of eigenvalues and eigen-
vectors is the basis for principal component analysis.