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Eigen Values and Eigen Vectors Eigen values and Eigen vectors are important in many areas of nu- merical computation and engineering applications. Deﬁned as the zeros of the characteristic polynomial: f (λ)= |A - λI| Useful in analysis of convergence characteristics of iterative meth- ods Ratios of Eigen values (largest to smallest) is a measure of the con- dition of a matrix. Applications include solutions of differential equations relating to physical characteristics of a structure(such as principal stress, mo- ments of inertia, vibration analysis, etc. ITCS 4133/5133: Numerical Comp. Methods 1 Regression

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### Transcript of Eigen Values and Eigen Vectors - UNC Charlotte FAQ · Eigen Values and Eigen Vectors Eigen values...

Eigen Values and Eigen Vectors

� Eigen values and Eigen vectors are important in many areas of nu-merical computation and engineering applications.

� Defined as the zeros of the characteristic polynomial:

f (λ) = |A− λI|

� Useful in analysis of convergence characteristics of iterative meth-ods

� Ratios of Eigen values (largest to smallest) is a measure of the con-dition of a matrix.

� Applications include solutions of differential equations relating tophysical characteristics of a structure(such as principal stress, mo-ments of inertia, vibration analysis, etc.

ITCS 4133/5133: Numerical Comp. Methods 1 Regression

Application

ITCS 4133/5133: Numerical Comp. Methods 2 Regression

Application

ITCS 4133/5133: Numerical Comp. Methods 3 Regression

Determining Eigen Values: Power Method

� An iterative procedure for determining the largest eigen value

� Begin with the definition,

λx = Ax

� Assume an initial guess for z, thus

w = Az

� If z is the eigen vector, then zk = wk, else iterate.

� Each iteration, z is scaled by its largest component.

wk ≈ λzk =⇒ λ ≈wk

zk

� Given that z is scaled, λ ≈ wk

ITCS 4133/5133: Numerical Comp. Methods 4 Regression

Power Method: Algorithm

ITCS 4133/5133: Numerical Comp. Methods 5 Regression

Power Method: Example

ITCS 4133/5133: Numerical Comp. Methods 6 Regression

Accelerated Power Method

� For symmetric matrices, we can use the Rayleigh coefficient for thelambda estimate to accelerate the convergence,

λ =zTw

zTz

ITCS 4133/5133: Numerical Comp. Methods 7 Regression

Shifted Power Method

� How can we determine the other Eigen values?

� Property: If λ1, · · · , λn are the eigen values of A, then the eigenvalues of A−bI are µ1 = λ1− b, · · · , λn− b. Eigen vectors are thesame.

� If we know the eigen value λ of a A, then a second eigen value canbe found by applying the power method to the shifted matrix A asfollows:

B = A− bI

ITCS 4133/5133: Numerical Comp. Methods 8 Regression

Shifted Power Method:Example

ITCS 4133/5133: Numerical Comp. Methods 9 Regression

Determining Eigen Values: Inverse PowerMethod

� To compute the smallest Eigen value of a matrix.

� Apply power method to A−1.

� Compute the reciprocals of the Eigen values of A−1; the dominantEigen value is the smallest Eigen value of A.

� In practice, the inverse of A is not computed.

A−1z = w =⇒ Aw = z

ITCS 4133/5133: Numerical Comp. Methods 10 Regression

Inverse Power Method:Algorithm

ITCS 4133/5133: Numerical Comp. Methods 11 Regression

Inverse Power Method:Example

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ITCS 4133/5133: Numerical Comp. Methods 13 Regression