1f_eigenproblemoverview

7/29/2019 1f_EigenProblemOverview

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Eigen problemsRevision



• Mathematics: Solve systems of differentialequations

• Vibration analysis: Describes frequency and

mode of vibration respectively• Mechanics: Represent principal stresses and

the principal axes of stress in bodies

•Digital signal processing: Use in thetheoretical analysis of adaptive filteralgorithms

Used in …

TUT: EL Voges 2



Outcomes

• Write down the characteristic equation for a

given matrix A

• Calculate the eigenvalues of A

•

Calculate the eigenvectors associated withdistinct and repeated eigenvalues

TUT: EL Voges 3



• Seeking non trivial solutions to the equation

• A is a square matrix

• x is an eigenvector of A (a non zero vector)

• is an eigenvalue of A corresponding to x

Eigenvalue problem

Ax x

TUT: EL Voges 4



Eigenvectors of a square matrix: non- zerovectors that, when multiplied by the matrix,

remain parallel to the original vector

Changes length ,not direction

What is an eigenvector?

Ax x

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An eigenvalue is the amount by which the

eigenvector is scaled when multiplied by the

matrix

• Stretching:

• Shrinking:

•

Reversal of direction :

What is an eigenvalue?

0

0 1

1

TUT: EL Voges 6

http://localhost/var/www/apps/conversion/tmp/scratch_6//upload.wikimedia.org/wikipedia/commons/5/58/Eigenvalue_equation.svg








What is an “eigenvalue”

Determines the amount the eigenvector is scaled under

the linear transformation

• Eigenvalue of +2: eigenvector is doubled in

length and points in the same direction

• Eigenvalue of −1: eigenvector is reversed in

direction

TUT: EL Voges 7



Practical example

0< <1

>1

<0

TUT: EL Voges 8



http://en.wikipedia.org/wiki/Eige

nvalue

Idea from wikipedia

http://en.wikipedia.org/wiki/Eigenvalue






Physical example (wikipedia)

TUT: EL Voges 10



Transformations in a plane along

with their 2×2 matrices, eigenvalues,

and eigenvectors (Wikipedia)

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12

Horizontal shear Scaling Unequal scaling Rotation by φ

illustration

matrix

characteristicequation

(1 − λ)2 = 0 (λ − k )2 = 0 (λ − k 1)(λ − k 2) =0

λ2 − 2λ cos φ + 1= 0

eigenvalues λi λ1=1 λ1=k λ1 = k 1, λ2 = k 2

λ1,2 = cos φ ± i

sin φ = e ± i φ

eigenvectors 1u 1, 0

T

1

2

u 1, 0

u 0,1

T

T

1

2

u 1,

u 1,

T

T

j

j

1

2

u 1, 0

u 0,1

T

T



The mathematics

• A is a square n × n matrix. Calculate

– Eigenvalue from the characteristic equation

– Eigenvector X from the matrix equation

0 A I

0 A I X

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• Sum of the eigenvalues of A = trace A

•

Product of the eigenvalues of A = det(A)

• Eigenvalues of

• Eigenvalues of , etc.

Properties if eigenvalues (p28)

1 1A

i

Ai

k k

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Calculate the eigenvalues and

corresponding eigenvectors



Example 1

3 2

2 3 A

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Example 2

1 0 11 2 1

2 2 3

A

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Example 3

6 5

5 4 B

TUT: EL Voges 18



Example 4

2 2 11 3 1

1 2 2

B

TUT: EL Voges 19



Let's practice!

Worksheet 2

1f_eigenproblemoverview

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