| course |
course_year |
question_number |
tags |
title |
year |
Numerical Analysis |
IB |
60 |
IB |
2018 |
Numerical Analysis |
|
Paper 1, Section II, D |
2018 |
Show that if $\mathbf{u} \in \mathbb{R}^{m} \backslash{\mathbf{0}}$ then the $m \times m$ matrix transformation
$$H_{\mathbf{u}}=I-2 \frac{\mathbf{u} \mathbf{u}^{\top}}{|\mathbf{u}|^{2}}$$
is orthogonal. Show further that, for any two vectors $\mathbf{a}, \mathbf{b} \in \mathbb{R}^{m}$ of equal length,
$$H_{\mathbf{a}-\mathbf{b}} \mathbf{a}=\mathbf{b} .$$
Explain how to use such transformations to convert an $m \times n$ matrix $A$ with $m \geqslant n$ into the form $A=Q R$, where $Q$ is an orthogonal matrix and $R$ is an upper-triangular matrix, and illustrate the method using the matrix
$$A=\left[\begin{array}{rrr}
1 & -1 & 4 \\
1 & 4 & -2 \\
1 & 4 & 2 \\
1 & -1 & 0
\end{array}\right]$$