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I'm preparing or my exam in linear algebra and I'm stuck with a question. I've tried to find some help in my textbook (Linear Algebra and its applications, 4th Edition, By David C. Lay). I can't find anything about it (maybe because the question is written in danish and I'm having trouble translating it right?).

I'm asked to find the limiting value, $\lim_{n \to \infty}A^nx$ where

$$ A = \begin{bmatrix} 0.25 & -0.75 & 0.75 \\[0.3em] -0.5 & 0 & 0.5 \\[0.3em] 0.25 & -0.25 & 0.75 \end{bmatrix} $$ $$ x = \begin{bmatrix} 2 \\[0.3em] 3 \\[0.3em] 3 \end{bmatrix} $$

How am I supposed to solve this? I'm not asking you to calculate the answer for me, but I'm asking for the right way to solve this kind of problem.

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    You probably mean $n \rightarrow \infty$.2012-08-20
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    Yes of course. That's a typo. Thanks! (corrected in question)2012-08-20
  • 1
    Do you know the Jordan Normal form of a matrix? If yes: what happens if you calculate a power of $BAB^{-1}$2012-08-20
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    I know the $A^k = PD^{k}P^{-1}$, but the limit confuses me.2012-08-20
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    Matrix multiplikation is continuos. If $X= \lim D^k$ exists, you can drag the limit into the product.2012-08-20

2 Answers 2

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$A$ is clearly diagonalisable, as the eigenvalues of $A$ are $-1/2,1/2,1$. Therefore, $A = PDP^{-1}$, where

$$D = \begin{bmatrix} -0.5 & 0 & 0 \\[0.3em] 0 & 0.5 & 0 \\[0.3em] 0 & 0 & 1 \end{bmatrix}$$ Now, $A^n = PD^nP^{-1}$, therefore, $\displaystyle \lim_{n \rightarrow \infty} A^n = \lim_{n \rightarrow \infty} PD^nP^{-1} = P X P^{-1}$, where, $$X = \begin{bmatrix} 0 & 0 & 0 \\[0.3em] 0 & 0 & 0 \\[0.3em] 0 & 0 & 1 \end{bmatrix}$$ Now, all that remains is to find $P,P^{-1}$. I'll leave this easy calculation, as the method is clear.

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You can see that the sup norm of matrix A, i.e., $\max\{a_{ij}\}$, is 3/4. Then $A:\mathbb{R}^n \rightarrow \mathbb{R}^n$ is a contraction. Since $R^n$ is a complete metric space there is a unique $y\in \mathbb{R}^n$ such that $Ay=y$ and we have $y=\lim_{n\to \infty}A^nz$, for every $z\in \mathbb{R}^n$, in particular for the given $x$. So, find y such that $Ay=y$. This limit do not depends of the norm because all norms in $\mathbb{R}^n$ are equivalents.

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    @ Euler 88 , sup norm is not a submultiplicative norm; moreover $A$ is not a contraction because $1$ is one of its eigenvalues.2015-07-05
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    Yes, you're right..2015-07-05