Let $A$ be a $n \times m$ matrix with real entries, and let $B = A A ^ { t }$ and let $\alpha$ be the supremum of $x ^ { t } B x$ where supremum is taken over all vectors $x \in \mathbb { R } ^ { n }$ with norm less than or equal to 1. Consider
$$C _ { k } = I + \sum _ { j = 1 } ^ { k } B ^ { j }$$
Show that the sequence of matrices $C _ { k }$ converges if and only if $\alpha < 1$.