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Matrix Theory, (Fall 1990) \\[.2in]
Assignment 7\\[.2in]
PART II \\
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Due: Mon., Oct. 15

$\left<\left<\right.\right.$Householder reflections$\left.\left.\right>\right>$\\
\begin{enumerate}
\item If {\em w} is a unit vector show that the matrix 
${\em Q} = {\em I} - 2{\em ww}^T$ is orthogonal.

\item Describe geometrically the action of ${\em Q}$ on an
arbitrary vector ${\em u} \in {\em R}^n$. (Note that the transformation
is a reflection about the median of an appropriately defined isosceles
triangle).

\item Given a vector ${\em a}_1 \in  {\em R}^n$ use the results in (1)--(2)
to construct an orthogonal matrix ${\em Q}$ that maps ${\em a}_1$ int a vector
${\em b}_1$ whose only nonzero component is the first, $b_{11}$. \\
 (note: since $|b_{11}|^2 = \sum a_{1i}^2$, chose $b_{11} = -\frac{a_{11}}
{|a_{11}|}\parallel {\em a}_1\parallel$ ).\\
That is, show that
\[ w_{11} = -\frac{a_{11}}{\parallel a_{11}\parallel}
\left(\frac{\left|a_{11}\right|+\parallel{\em a_1}\parallel}{S}\right) \]
\[ w_{1i} = -\frac{a_{1i}}{S} \ \ \ ,\ \ \ i\neq 1 ,\]
with $S^2 = 2\parallel{\em a}_1\parallel(\left|a_{11}\right|+
\parallel{\em a}_1\parallel)$.
\\
(you must show that $S^2 = \parallel {\em b}_1 - {\em a}_1\parallel $ ).

\item Also, construct an orthogonal matrix ${\em Q}_l$ that transforms a 
vector ${\em a}_l$ into a vector ${\em b}_l$ such that 
\begin{eqnarray*}
b_{li} & = & a_{li}\ \ \ ,\ \ \ i = 1,\cdots , l-1 \\
b_{li} & = & 0\ \ \ ,\ \ \ i = l+1,\cdots , n.
\end{eqnarray*}
What must $b_{ll}$ be? Choose its sign in accordance to your
choice for $b_{11}$.

\item Utilizing the above ideas, describe an algorithm that accomplishes a 
factorization ${\em A = QR}$ where ${\em A}$ is an $m\times n$ ($m\geq n$)
matrix of rank n.

\item How many operations does the algorithm in (6) require?

\item For the same matrix ${\em A}$ estimate the operation count
for a ${\em QR}$ factorization using the Gram--Schmidt procedure.
Compare the two methods.
\end{enumerate}
