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In some linear algebra books/blogs, I have found that a power method can be done only on diagonalizable matrix with a dominant eigenvalue. In other books/blogs, I have found that a irreducible, stochastic, aperiodic matrix can converge on the power method. I am working with irreducible, stochastic, aperiodic and it may be diagonalizable sometime or not sometimes. I want to know if it would always converge on the power iteration method. I also want to know why a non-diagonal matrix with a dominant eigenvalue can not converge?

Bernard
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criticalmind
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  • Of course the power method can be "done" on any matrix, and the issue is whether it gives useful information about "a dominant eigenvalue." But I think I don't quite know the crux of your problem. If $M$ is stochastic, its largest eigenvalue is 1. – hardmath Sep 13 '19 at 20:43
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    You might want to check the Perron-Frobenius theory, a matrix with positive or null entries has its largest eigenvalue positive and real with a positive eigenvector. This is very compatible with the power method. – Lutz Lehmann Sep 13 '19 at 20:44
  • I know, But why on most of the blogs it is said that the matrix should be diagonalizable for power method to work? I don't get that. Even in wikipedia, the blog "Power iteration" states same. @LutzL – criticalmind Sep 13 '19 at 21:02
  • It is a reasonable Question, but it would be improved by pointing out a blog where "it is said that the matrix should be diagonalizable". I think this might be the least important of the matrix properties listed (for using the power method), and it would help to identify whether convergence of the eigenvector or of the eigenvalue is important to you. – hardmath Sep 13 '19 at 21:08
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    I don't read the Wikipedia article "Power iteration" to say the matrix "should be diagonalizable for the power method to work." Indeed see further down in the article for application of the power method to non-diagonalizable matrices. What is true is that the analysis is easier for diagonalizable matrices, and frequently the algorithm is applied to such matrices (e.g. real symmetric cases). I read your Question to be mainly interested in stochastic matrices, but on closer study I can see I might be mistaken about that. – hardmath Sep 13 '19 at 21:20

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The power method can be used to analyze matrices that are not diagonalizable. The way to do it is using a similar transformation, called Jordan decomposition (see here), where unlike the eigenvalue decomposition, Jordan decomposition always exists.

Behrouz
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