# Subsection30.1Exercises

##### Exercise30.1

Use that the square matrix $A$ has a $QR$ factorisation to prove that

\begin{equation*} |\det(A)| \leq \prod_{j=1}^m \|a_j\|_2, \end{equation*}

where, as usual, $a_j$ is column $j$ of $A\text{.}$

##### Exercise30.2

Let $F$ be a Householder reflector. That is, for some vector $v\text{,}$

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

Determine the eigenvalues, determinant, and singular values of $F\text{.}$