# linear algebra – Prove that this sequence of matrices converges iff \$alpha

This question was part of my previous year mid term exam of Linear Algbera and I am really struck on it.

Let A be $$n times m$$ matrix with real entries, and let $$B=AA^t$$ and let $$alpha$$ be the supremum of $$x^t Bx$$ where supremum is taken over all vectors $$xin mathbb{R}^n$$ 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 iff $$alpha <1$$.

The problem is that I am not even able to start.

I have been following Hoffman and Kunze linear algebra and I am doing a graduate level course in it.

Any help would be deeply appreciated.