# Linear algebra/numerical analysis: Power method question(s)

• fluidistic
In summary, the conversation discusses finding an approximate value for the largest eigenvalue of a given matrix A using the power method or the normalized one, and using Rayleigh's quotient. However, it is noted that Rayleigh's quotient can only be used for Hermitian matrices. The conversation then questions whether Rayleigh's quotient can still be used for non-Hermitian matrices and if there is an alternative method involving an infinity norm. The conversation also mentions that although the example in the notes is not a symmetric matrix, the method should still work.
fluidistic
Gold Member

## Homework Statement

Ok I understand how to find an approximate value for the largest eigenvalue of a given matrix A. I use the power method (or the normalized one) to find an eigenvector associated to the approximate largest (in the sense that its modulus is the largest) eigenvalue.
However looking in wikipedia about Rayleigh's quotient, it seems that the matrix A must be Hermitian (see http://en.wikipedia.org/wiki/Rayleigh_quotient).

In my assignment I'm been given a matrix A which is NOT Hermitian! So I wonder 2 things I'd like an anwer:
1)Can I still use Rayleigh's quotient, why or why not?
2)I'm more than sure there's another way to get the greatest eigenvalue from the approximate eigenvector, that does not involve Rayleigh's quotient but does involve an infinity norm. I've searched on the web about this and found really nothing. If you know it, please let me know what is this alternative.

the example in your notes is not a symmetric matrix - Ex3

if you think about it geometrically, it should still work - each time you multiple by the matrix, the component in the direction of the dominant eigenvector gets scaled the most

the thing about hermitian matricies, are that they're guaranteed to have an orthonormal basis of eigenvectors, so the method above should converge pretty quickly

lanedance said:
the example in your notes is not a symmetric matrix - Ex3

if you think about it geometrically, it should still work - each time you multiple by the matrix, the component in the direction of the dominant eigenvector gets scaled the most

the thing about hermitian matricies, are that they're guaranteed to have an orthonormal basis of eigenvectors, so the method above should converge pretty quickly

If someone could help me with part 2) I'd be glad too.

## 1. What is the power method in linear algebra/numerical analysis?

The power method is an iterative algorithm used to find the eigenvalues and eigenvectors of a square matrix. It is commonly used to find the dominant eigenvalue and corresponding eigenvector, which can be useful in applications such as principal component analysis and Google's PageRank algorithm.

## 2. How does the power method work?

The power method involves repeatedly multiplying the initial vector by the given matrix, and then normalizing the resulting vector. This process is repeated until the vector converges to the dominant eigenvector, which corresponds to the dominant eigenvalue.

## 3. What is the convergence rate of the power method?

The convergence rate of the power method depends on the ratio between the dominant eigenvalue and the next largest eigenvalue. If this ratio is large, the power method will converge quickly. However, if the ratio is small, the convergence rate will be slower.

## 4. Can the power method be used for non-square matrices?

No, the power method can only be used for square matrices. For non-square matrices, other methods such as the singular value decomposition (SVD) can be used to find the dominant singular value and corresponding singular vector.

## 5. What are the limitations of the power method?

The power method may not converge if the matrix has repeated eigenvalues or if the dominant eigenvalue has a low multiplicity. In addition, the method may also be sensitive to the initial vector chosen, and may not converge to the desired eigenvector if the initial vector is not in the direction of the dominant eigenvector.

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