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Eigen Value Problems-Machine Learning and MATLAB-Lecture Slides, Slides of Machine Learning

Alliance UniversityMachine Learning

Dr. Sahaujas Advait delivered this lecture at Alliance University for Machine Learning and MATLAB course. Its main points are: Eigenvalue, Problems, MATLAB, Implementaion, Characteristics, Polynomial, Companion, Matrix, Jordan, Schur, Form

Typology: Slides

2011/2012

Uploaded on 07/17/2012

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Chapter 4
Eigenvalue Problems
(MATLAB Implementation)
Dr. Muhammad Hanif Durad
Department of Computer and Information Sciences
Pakistan Institute Engineering and Applied Sciences
[email protected]
Most of the codes have been adopted from various books for
Numerical Computing using MATLAB their references have been
given in short on each topic
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Chapter 4

Eigenvalue Problems(MATLAB Implementation)

Dr. Muhammad Hanif Durad Department of Computer and Information SciencesPakistan Institute Engineering and Applied Sciences

[email protected]

Most of the codes have been adopted from various books forNumerical Computing using MATLAB their references have beengiven in short on each topic

docsity.com

Dr. Hanif Durad

Lecture Outline (1/2)  Characteristics Polynomial  Companion Matrix  Algebraic Multiplicity  Gerschgorin Theorem  Bauer-Fike Theorem  Eigenvalue Sensitivity  Jordan Form  Schur Form

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Characteristic Polynomial>> A=[3 1;1 3]A = (^3 11 3) >> poly(A)ans = 1 -^

Dr. Hanif Durad

(^4) docsity.com

Companion Matrix

>> [V,D] = eig(B)V =0.

D =^4

Dr. Hanif Durad

>> c=[1 -6 8]c =^1

-6^8

>> B=compan(c)B =^6

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Algebraic-Geometric Multiplicity(2/2)

Dr. Hanif Durad

V =^1

-^

D =^1

When^ λ=1; AM=2; G.M=1When^ λ=0; AM=1; G.M=

v1^ v

v3 (^) λ 1 λ λ 2 3

v1 = v^2

Defective Matrix (?)Not LinearlyIndependent

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Gerschgorin Theorem>> A=[4 2 3;-2 -5 8;1 0 3]A = (^4 2 3) -2 -5^81 0 3 >> C = Gershg(A)C = (^4 0 5) -5 0 10 3 0 1

Dr. Hanif Durad radius

center

│z-4│=5^ │z+5│=10 │z-3│=

-15≤^

λ^ ≥^9 ??

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Gerschgorin Theorem

Dr. Hanif Durad

(^10) docsity.com

Bauer-Fike Theorem (1/2)>> A=[3 1;0 2]>> [X,D] = eig(A)X =1.0000^ -0.7071 0 0.7071D = 3 0 0 2

Dr. Hanif Durad

E=[0 0; 10^-4 0];>> cond(X,2)ans =2.4142>> norm(E,2)ans =1.0000e-

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Eigenvalue Sensitivity-Example 4.7 (1/5)>> A=gallery(3)A = -149^ -50 -154 537 180 546 -27^ -9^ -

Dr. Hanif Durad

[X,D] = eig(A)

%left eigenvector X =0.

-0.^
0.^
-0.^
0.^
D =1.

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Eigenvalue Sensitivity-Example 4.7 (2/5)

[Y,D] = eig(A.') %right eigenvectorY =0.

0.^
-0.^
0.^
-0.^
D =1.

Dr. Hanif Durad

c = condeig(A)c =603.6390395.2366219.2920>> 1./cans =0.00170.00250.

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Eigenvalue Sensitivity-Example 4.7 (4/5)>> C=AC =-149^ -50 -154 537 180

-27^ -^
>> C(2,2)=179.99C =-149.0000 -50.0000 -154.0000537.0000 179.9900 546.0000-27.

Dr. Hanif Durad

(^16) docsity.com

Eigenvalue Sensitivity-Example 4.7 (5/5)

Dr. Hanif Durad

[X2,D2] = eig(C)X2 =-0.2998 - 0.0414i

-0.2998 + 0.0414i

0.^
0.^

-0.0160 + 0.0426i

-0.0160 - 0.0426i

D2 =1.6642 + 1.0543i

1.6642 - 1.0543i

2.6616 docsity.com

Schur Form>> schur(A)ans =^1 10

Dr. Hanif Durad

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Triangular Form>> A=[1 3 -1 0; 0 2 -6 1; 0 0 -3 2;0 0 0 4];>> [X,D] = eig(A)X =1.0000^ 0.

Dr. Hanif Durad

D =^1
0 0 0 2 0 0 0 0 -3^00 0 0

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