Prove that a serie of matrices converges to $0$

Question

I've been struggling with this problem for a while I cannot find the solution, hope you can help me.

Prove that the series $(I + A + A^2 + ...)$ converges if $\begin{Vmatrix}B\end{Vmatrix} < 1$, where $B = PAP^{-1}$. What is the implication of this result?, Construct a simple example to see usefulness of the result in practical computations.

Answer 1

Couple of tips which probably has nothing to do with your problem or may be it has

• From real analysis, for any variable $x$ with $|x|<1$, $(1-x)^{-1}=1+x+x^2\dots$. This is not directly extendable to matrices though.
• $||A||_2=\sigma_1(A)=\sqrt{\lambda_{max}(A^TA)}$
• $\det(PQP^{-1}-\lambda I)=\det(Q-\lambda I)$ for any invertible $P$ and matrix $Q$.
• If $B= PAP^{-1}$, then $B^n = PA^{n}P^{-1}$