Eigenvalue Problems part 1-Parallel Computing-Lecture Slides, Slides of Parallel Computing and Programming

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Lecture Outline



Eigenvalue Problems



Existence, Uniqueness and Conditioning



Computing Eigenvalues and Eigenvectors

origins (1/2)Eigenvalue Problems. Their



Structural Engineering [K

x

2

2

M

x

]



mechanical system,..]Stability analysis [e.g., electrical networks,



Bifurcation analysis [e.g. in fluid flow]



equation..]Electronic structure calculations [Schrödinger



world-wide web.Application of new era: page ranking on the

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LecN8_R.pdf,P-

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Eigenvalues and Eigenvectors

Note:  may be complex even if A is real

Eigen_002.pdf, P-

Geometric Interpretation

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Characteristic Polynomial (1/2)

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PolynomialExample: Characteristic

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Characteristic Polynomial (2/2)

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Assignment -

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Find the characteristic polynomial, eigenvalues

and eigen vectors of the matrix

A

AEM Alan Jeffrey.pdf, P-

DiagonalizabilityMultiplicity and

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SubspacesEigenspaces and Invariant

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(1/2)Examples: Matrix Properties

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(II)

(I)

III

(a)

(I

V

(a)

(b) (b)

(2/2)Examples: Matrix Properties

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V

I)

(b) (a)

V

(a) (b) (c)

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20

Consider the following example.

4

2

3

2

5

8

1

0

3

A























C The circles that bound the Eigenvalues are:

1 : Center point (4,0) with radius r

1

= |2|+|3|=

C

2 : Center point (-5,0) with radius r

2 =|-2|+|8|=

C

3 : Center Point (3,0) with radius r

3 =|1|+|0|=

5

10

-15 -

• 5

10

Union of the Circles

5

-15 -

• 5 10

Eigenvalueslocation of thethe actualthe right mark The red dots to

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1

0

7

2

5

0

4

4

3

A























C The circles that bound the Eigenvalues are:

1 : Center point (1,0) with radius r

1

= |0|+|7|=

C

2 : Center point (-5,0) with radius r

2 =|2|+|0|=

C

3 : Center Point (-3,0) with radius r

3 =|4|+|4|=

5

10

-15 -

• 5 10

inside the union of all the circles. We see all the Eigenvalues lie

Consider another example.