Eigen-analysis and the Power Method

Module Goals Power Method Shift technique (optional) Inverse Method Accelerated Power Method

Power method The special advantage of the power method is that the eigenvector corresponds to the dominant eigenvalue and is generated at the same time. The inverse power method solves for the minimal eigenvalue/vector pair. The disadvantage is that the method only supplies obtains one eigenvalue

Power Method Readers Digest Version Eigenvalues can be ordered in magnitude and the largest is called the dominant eigenvalue or spectral radius. Think about how eigenvalues are a reflection of the nature of a matrix. Now if we multiply by that matrix over and over again..eventually the biggest eigenvalue will make everyone else have eigen-envy. One λ to rule them all, One λ to find them, One λ to bring them all and in the darkness bind them.

Power Method In general continue the multiplication: where,

Power Method Factor the large l value term As you continue to multiply the vector by [A]

Power Method The basic computation of the power method is summarized as

Power Method The basic computation of the power method is summarized as The equation can be written as:

The Power Method Algorithm (algorithm 3.3.1 pg 107) y=nonzero random vector Initialize x = A*y vector for k =1,2,…n y=x/||x|| x =Ay (x is the approximate eigenvector) approximate eigenvalue μ= (yT*x)/(yT*y) r=μy-x k++

Example of Power Method Consider the follow matrix A Assume an arbitrary vector x0 = { 1 1 1}T

Example of Power Method Multiply the matrix by the matrix [A] by {x} Normalize the result of the product

Example of Power Method

Example of Power Method As you continue to multiple each successive vector l = 4 and the vector uk={1 0 0}T

Shift method (optional) It is possible to obtain another eigenvalue from the set equations by using a technique known as shifting the matrix. Subtract the a vector from each side, thereby changing the maximum eigenvalue

Shift method The eigenvalue, s, is the maximum value of the matrix A. The matrix is rewritten in a form. Use the Power method to obtain the largest eigenvalue of [B].

Example of Power Method Consider the follow matrix A Assume an arbitrary vector x0 = { 1 1 1}T

Example of Power Method Multiply the matrix by the matrix [A] by {x} Normalize the result of the product

Example of Power Method Continue with the iteration and the final value is l = -5. However, to get the true you need to shift back by:

Inverse Power Method The inverse method is similar to the power method, except that it finds the smallest eigenvalue. Using the following technique.

Inverse Power Method The algorithm is the same as the Power method and the “eigenvector” is not the eigenvector for the smallest eigenvalue. To obtain the smallest eigenvalue from the power method.

Inverse Power Method The inverse method is similar to the power method, except that it finds the smallest eigenvalue. Using the following technique.

Inverse Power Method The algorithm is the same as the Power method and the “eigenvector” is not the eigenvector for the smallest eigenvalue. To obtain the smallest eigenvalue from the power method.

Inverse Power Method The inverse algorithm use the technique avoids calculating the inverse matrix and uses a LU decomposition to find the {x} vector.

Example The matrix is defined as:

Accelerated Power Method The Power method can be accelerated by using the Rayleigh Quotient instead of the largest wk value. The Rayeigh Quotient is defined as:

Accelerated Power Method The values of the next z term is defined as: The Power method is adapted to use the new value.

Example of Accelerated Power Method Consider the follow matrix A Assume an arbitrary vector x0 = { 1 1 1}T

Example of Power Method Multiply the matrix by the matrix [A] by {x}

Example of Accelerated Power Method Multiply the matrix by the matrix [A] by {x}

Example of Accelerated Power Method

Example of Accelerated Power Method And so on ...