# Subsection31.1Exercises

##### Exercise31.1

Show that computing the matrix product $B=FA\text{,}$ where $F$ and $A$ are both $m\times m$ matrices, using the “usual” algorithm, takes $\mathcal{O}(m^3)$ operations.

Let $F$ be the Householder reflector

\begin{equation*} F = I - 2 {v v^\star}, \end{equation*}

where $v$ is an $m$-vector such that $\|v\|_2=1\text{.}$ Show that computing $B=FA$ is the same as $B = A-2v (v^\star A)\text{.}$ How many operations would this require?

##### Exercise31.2

Use the existance of the Schur Form of the matrix $A \in \Cmm$ to prove that $\lim_{n \to \infty} \|A^n\| = 0 \iff \rho(A) \lt 1,$ where $\rho(A)$ is the spectral radius of $A\text{.}$