# Eigen Values and Eigen Vectors - UNC Charlotte FAQ · PDF file Eigen Values and Eigen Vectors Eigen values and Eigen vectors are important in many areas of nu-merical computation and

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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

• 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 the lambda 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 eigen values of A−bI are µ1 = λ1− b, · · · , λn− b. Eigen vectors are the same.

� If we know the eigen value λ of a A, then a second eigen value can be found by applying the power method to the shifted matrix A as follows:

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 Power Method

� 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 dominant Eigen 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

ITCS 4133/5133: Numerical Comp. Methods 12 Regression

• ITCS 4133/5133: Numerical Comp. Methods 13 Regression

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