Problem 294 of 333HardLinear AlgebraP294

How far a shear stretches

Let
$A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$
- What are the eigenvalues of $A$ ?
- What is $\|A\|_2$ , the largest factor by which $A$ stretches a unit vector, and which unit vector achieves it? What is the condition number of $A$ ?
- Answer the same for $A^{10}$ , and describe how $\|A^n\|_2$ grows with $n$ .
Why do the eigenvalues give the wrong picture here?
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