Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

The question is from an exercise in Gilbert Strang's Linear Algebra and its Applications.

The powers $A^k$ approach zero if all $|\lambda_i|<1$, and they blow up if any $|\lambda_i|>1$. Peter Lax gives four striking examples in his book Linear Algebra. $$A = \left( \begin{array}{cc} 3& 2 \\ 1& 4 \\ \end{array} \right)\qquad B = \left( \begin{array}{cc} 3 & 2 \\ -5 & -3 \\ \end{array} \right)\qquad C = \left( \begin{array}{cc} 5& 7 \\ -3& -4 \\ \end{array} \right)\qquad D = \left( \begin{array}{cc} 5& 6.9 \\ -3& -4 \\ \end{array} \right)$$ $$\|A^{1024}\|>10^{700}\qquad B^{1024}=I\qquad C^{1024}=-C\qquad \|D^{1024}\|<10^{-78}$$ Find the eigenvalues $\lambda=e^{i\theta}$ of $B$ and $C$ to show that $B^4=I$ and $C^3=-I$.

Here is my question:

Why are these examples so special? Is it because that all of them contain the number "1024"? Or such examples are hard to construct?

share|improve this question
1  
This sounds pretty subjective. I don't see anything too striking about them at all, other than maybe the fact that all components are obviously $>1$ in magnitude yet the powers of the middle two are kept (relatively) stable while the last shrinks. –  anon Aug 10 '11 at 3:16
5  
As I see, one gets $D$ by a "small" perturbation on $C$. –  Jack Aug 10 '11 at 3:21

2 Answers 2

up vote 16 down vote accepted

There is nothing particularly special about $1024$; it just happens to be convenient to compute matrices to $2^n$ powers because you can repeatedly square.

What's supposed to be striking (of course this is subjective) is that $A, B, C, D$ all have small, superficially similar-looking entries, but after sufficient iteration have very different qualitative behavior; moreover, if you didn't know a lot of linear algebra, you would be hard-pressed to guess which matrices would admit which behavior just by looking at them.

Examples $A, D$ are not hard to construct in the sense that a random matrix you choose will do one of those two things. Examples $B, C$ are more fine-tuned, but not hard to construct by hand once you understand how to construct matrices with a given characteristic polynomial.

share|improve this answer
    
How to choose the random matrix? –  Jack Aug 10 '11 at 17:36
1  
Any way you want. Just write down four small integers. (If you're asking how to make a precise statement, note that periodic behavior is impossible unless the determinant is a root of unity, and the set of matrices with this property has measure zero in the space of all matrices.) –  Qiaochu Yuan Aug 10 '11 at 17:44

The examples display three different behaviors:

    1. $\|A^k\|\to\infty$
    2. $B^k$ and $C^k$ cycle with periods of $4$ and $6$ respectively, always with norm $1$
    3.$\|D^k\|\to0$

$1024$ is just a large number that displays these behaviors strikingly. Since $1024=0\pmod{4}$ and $1024=4\pmod{6}$, so $B^{1024}=I$ and $C^{1024}=-C$.

Peter Lax said the examples were "striking", not "special".

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.