Skip to main content
\(\newcommand{\Cs}{\mathbb{Cs}} \newcommand{\Cm}{\mathbb{C}^{m}} \newcommand{\Cmm}{\mathbb{C}^{m\times m}} \newcommand{\Cmn}{\mathbb{C}^{m\times n}} \newcommand{\R}{\mathbb{R}} \newcommand{\Rm}{\mathbb{R}^{m}} \newcommand{\Rn}{\mathbb{R}^{n}} \newcommand{\Rmm}{\mathbb{R}^{m\times m}} \DeclareMathOperator{\Range}{range} \DeclareMathOperator{\Rank}{rank} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\Span}{span} \DeclareMathOperator{\range}{range} \DeclareMathOperator{\Null}{null} \newcommand{\definiteintegral}[4]{\int_{#1}^{#2}\,#3\,d#4} \newcommand{\indefiniteintegral}[2]{\int#1\,d#2} \graphicspath{ {images/} } \usepackage{amsmath} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)

Section30The \(QR\) factorisation (NM, 13 Nov)

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 \(F\text{.}\)