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https://math.stackexchange.com/ques...problem-from-eberhard-zeidlers-first-volume-o

I have a question from Eberhard Zeidler's book on Non-Linear Functional Analysis, question 1.5a, he gives as a reference for this question the book by Wilkinson called "The Algebraic Eigenvalue Problem", but he doesn't give a page where it appears and it's hard to find exactly the wording suitable for this question.

Anyway, the question is as follows:

Let $A$ be an $N\times N$ matrix with $N$ linearly independent eigenvectors $x_1,\ldots , x_N$ and corrseponding eigenvalues $\lambda_i$, where $|\lambda_1|>|\lambda_2| \ge |\lambda_3| \ge \ldots \ge |\lambda_N|$.

Let $x^{(0)}=\sum_{i=1}^N \alpha_i x_i , \ \alpha_1 \ne 0$, and $x^{(n)}=A^nx^{(0)}/\| A^n x^{(0)}\|_{\infty}$. Show that:

as $n\rightarrow \infty$, $x^{(n)}\rightarrow \frac{x_1}{\|x_1\|_{\infty}}$ and $\|Ax^{(n)}\|_{\infty} \rightarrow \lambda_1$ .

It appears on page 40.In the correspondence with the poster in the forum we didn't explicitly wrote the proof, he said it's too long; I still would like a full solution to this problem.

Or if you have a reference to this problem?

Thanks.