Eigen Values and Eigen Vectors - UNC Charlotte FAQ Eigen Values and Eigen Vectors Eigen values and...

download Eigen Values and Eigen Vectors - UNC Charlotte FAQ Eigen Values and Eigen Vectors Eigen values and Eigen

of 13

  • date post

    15-Mar-2020
  • Category

    Documents

  • view

    0
  • download

    0

Embed Size (px)

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

  • 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