Spectral Norm Proof

Question

I don't really understand the question below:

A matrix can have small eigenvalues but large spectral norm (i.e., largest singular value). Find such a matrix A of proper dimension so that all of its eigenvalues λ satisfy |λ| < 0.5, but its spectral norm satisfies ||A|| ≥ K for some K > 0 that can be made arbitrarily large.

I feel like I need to start with the definition of the spectral norm: $$||A||=\sqrt{\lambda_{\max}(A^TA)}$$

Not really sure if that's the right place to start or where to go from there. Any and all help is appreciated!

Thanks.

Answer 1

Start with $A=\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$ and find $\|A\| = 1$. Then $\| K A \| = |K|$. All eigenvalues are zero.