Let $A$ be a $3x3$ matrix which is not a diagonal matrix. Show that its eigenvalues are not all the same. Let $Q(x)$ be the corresponding quadratic form: show that $$\lim_{x\to 0} \frac{Q(x)}{||x||^2}$$
does not exist.
I have a solution to this problem, but it doesn't make much sense to me. I'm hoping someone here can produce something I can understand. Any help would be appreciated. Thanks!
$$A = \begin{pmatrix} 0& 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$$ is not diagonal but has only $0$ as eigenvalue. More generally every matrix of the form $$A = \begin{pmatrix} a& * & * \\ 0 & a & * \\ 0 & 0 & a \end{pmatrix}$$ (where $*$ is whatever you want) has only $a$ as eigenvalue but is not diagonal.