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?