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I have a symmetric real matrix $A$. Iteratively I am doing the following:

$0) A_1=A$

$1)$ Calculate the largest eigenvalue $\lambda_i$ and corresponding unit norm eigenvector $v_i$ (you may assume a single largest eigenvalue).

$2)$ Then $A_i$ is constructed from $A_{i-1}$ by setting the diagonal element $A_{i}(j,j)=\lambda_i (v_i(j))^2$.

Repeat until convergence i.e. $\lambda_{i+1}=\lambda_{i}$ for some $i$. The simulations $i$ did shows that this always converge, sometimes slower sometimes faster. Do anyone know why this procedure is converging, if so to what it is converging?

Thanks

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    $v_i$ is the unit eigenvector that corresponds to $\lambda_i$ which is the largest eigenvector. Probably you did not get the question right. – 2017-01-04
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    You can assume there is one largest eigenvalue, I hope this clarifies it. – 2017-01-04
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    Thanks. Please edit that into your question. – 2017-01-04

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