Linear Transformation Examples Matrix Eigenvalue problems

Md55
Eigenvalues and Eigenvectors
● Terminology:
Ax = λx matrix eigenvalue problem
A value of λλ for which x (≠ 0) is a solution eigenvalue,
(also known as characteristic value)
Solutions x (≠ 0) corresponding to a λ called eigenvectors
The set of eigenvectors is called the spectrum of A
Largest of absolute values of eigenvalues is spectral radius of A
● Determination of eigenvalues and eigenvectors:
Ax = λx = λIx, where I is the identity matrix
(A - λI)x = 0, homogeneous linear system with non-trivial solution
(x ≠ 0) if and only if D(λλ) = det(A - λI) = 0
Md56
Example
Ax = x A =
x
x
x
x x x
x x x
− ⎡
⎢
⎣
⎤
− ⎥ =
⎦
⎡
● Determination of eigenvalues :
5
λ ,
2
2
1 ⎤
, ⎢ ⎥,
so that
2 ⎣ 2 ⎦
− 5 1 + 2 2 = λ 1
2 1 − 2 2 = λ 2
− − x + x =
A− I x =
x + − − x =
D = A− I = −
( 5 λ)
1 2 2 0
( λ ) 0,
2 1 ( 2 λ)
2 0
and
5−λ ( λ) det( λ )
2
2
= 0
−2−λ Characteristic polynomial
2
D(
λ) = ( −5−λ)( −2−λ) − 4 = λ + 7λ + 6= 0
Whence, ( λ + 1)( λ + 6) = 0, so that λ = −1, − 6; i.e. λ =− 1, λ =−6.
● Determination of an eigenvector:
1 2
For
− + =
λ = λ = − :
− =
and these equations are linearly dependent
with a solution : = . This determines an eigenvector corresponding to λ = −
up to a scalar multiple. If we choose x 1 = , we obtain eigenvector x = e1
= .
⎡ ⎤ ⎡
⎢ ⎥ = ⎢
⎣ ⎦ ⎣
⎤
4x1 2x2 0
1 1
2x1 x2
0
x2 2x1 1 1
x1
1
1
x ⎥
2 2⎦