Unraveling the Mysteries of Power and Inverse Power Methods for Eigenvalues

In summary, the conversation discussed two exercises related to finding the largest and smallest eigenvalues of a matrix A using the power method and the inverse power method. The smallest and largest eigenvalues can be found by using the Rayleigh quotient of A and x. The eigenvalues of A are arranged in descending order and can be denoted as Ln, Ln-1, ..., L1. The convergence of x(k) in both exercises is of order 1 with a rate of convergence determined by the eigenvalues. Exercise 1 asked for the conditions that must be met for this to be true for each coordinate, while Exercise 2 assumed that A is symmetrical and x(k) converges towards a multiple of the eigenvector. In this
  • #1
pinodk
21
0
I have two excercises which have been causing me to tear my hair off for some time now.
(a) the power method to find largest eigenvalue of A is defined as x(k+1) = Ax(k)
(b) the inverse power method is to solve Ax(k+1) = x(k) to find smallest eigenvalue of A
(c) the smallest/largest eigenvalue is then found by the Rayleigh quotient of A and x : R_A(x) = (Ax,x)/(x,x)

The eigenvalues of A are arranged as lambda_n >= lambda_n-1 >=... lambda_2 >= lambda_1

Henceforth lambda_i will be written as Li, so
Ln >= Ln-1 ... >= L2 >= L1

Excercise 1.)
The convergence of x(k) in (a) and (b) is of order 1 with rate of convergence |Ln-1/Ln| and |L1/L2| respectively.
State conditions for this to be true for each coordinate.

that was the first, my question is then, what does that question mean (its translated from danish, hope i did it right :-S)? I read it as:
give the conditions for the eigenvalues that must be fulfilled in order for the above to be true, and show that it applies to each coordinate in x(k)...
If that makes sense to you, please help me on how to get on with this excercise, because I am still stucked...

Excercise 2.)
Assume that A is symmetrical, and that x(k) converges towards (a multiple of) the eigenvector, so that each coordinate of x(k) converges with rate of convergence c.
Show that L(k) converges by order 1 with rate of convergence c^2.

Again I am baffled... please get me started :-)
 
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  • #2
guess I am not the only one finding it difficult, eh? Or have i been unclear in the formulation?
 

1. What are power and inverse power methods for eigenvalues?

Power and inverse power methods are numerical algorithms used to find the eigenvalues of a square matrix. The power method involves repeatedly multiplying the matrix by a vector and taking the limit as the number of iterations approaches infinity. The inverse power method is similar, but instead uses the inverse of the matrix. Both methods can be used to find the dominant eigenvalue and corresponding eigenvector of a matrix.

2. How do power and inverse power methods work?

Both methods rely on the fact that the dominant eigenvalue of a matrix will have the largest absolute value. The power method starts with an initial vector and repeatedly multiplies it by the matrix, taking the limit as the number of iterations approaches infinity. This will eventually converge to the dominant eigenvector. The inverse power method, on the other hand, uses the inverse of the matrix to find the smallest eigenvalue, which is the inverse of the dominant eigenvalue.

3. What are the advantages of using power and inverse power methods?

Power and inverse power methods are relatively simple and efficient ways to find the dominant eigenvalue and eigenvector of a matrix. They are also useful for finding the smallest eigenvalue, which can be difficult to compute with other methods. Additionally, these methods can be applied to large matrices and can handle complex eigenvalues.

4. What are the limitations of power and inverse power methods?

One limitation of power and inverse power methods is that they only find one eigenvalue and corresponding eigenvector at a time. This means that multiple iterations are required to find all eigenvalues and eigenvectors of a matrix. Additionally, these methods may not converge for matrices with multiple eigenvalues of the same absolute value, and they may not work for matrices with zero eigenvalues.

5. How are power and inverse power methods used in real-world applications?

Power and inverse power methods have numerous applications in science and engineering. They are commonly used in physics, chemistry, and engineering to solve systems of differential equations and to analyze the stability of dynamical systems. These methods are also used in computer graphics, signal processing, and image processing for tasks such as image compression and noise reduction.

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