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{\large\bf C\&O 367, Winter 2001 \\ 
Assignment 2 
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{\large  Due on Thursday, Jan. 25, (at start of class)\\
 Instructor H. Wolkowicz
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\begin{enumerate}
\item
(10 marks) 
(Text: Problem 31, page 36) 
\begin{enumerate}
\item
Let $A$ be an $n \times n$ symmetric matrix. Diagonalize $A$ to show
that (the Raleigh quotient)
\[
\frac{x^tAx}{||x||^2}
\]
is greater than or equal to the smallest eigenvalue of $A$ for all $x
\neq 0$ in $\Re^n$.
\item
\label{small}
Show that the quadratic form $Q_A(x) = x^tAx$ is coercive if and only if
$A$ is positive definite.
\item Conclude from \ref{small} that if 
\[ f(x) = a + b^tx + \frac 12 x^t A x
\]
is any quadratic function where $a \in \Re,~ b \in \Re^n$ and $A$ is an
$n \times n$ symmetric matrix, then $f(x)$ is coercive if and only if
$A$ is positive definite.
\end{enumerate}

\item
(5 marks) 
Suppose that
\[ f(x) =  b^tx + \frac 12 x^t A x
\]
where $b \in \Re^n$ and $A$ is an $n \times n$ symmetric matrix.
Show that $f(x)$ is bounded below on $\Re^n$ if and only if the minimum
of f on $\Re^n$ is attained (i.e. there exists $\bar{x}$ such that
$f(\bar{x})=\min\limits_{x\in\Re^n} f(x)$).
\item
(5 marks)
(Text: Problems 1a and 1d, page 77) 
\item
(5 marks)
(Text: Problems 2b and 2c, page 77) 
\item
(5 marks)
(Text: Problems 9, page 78) 
\end{enumerate}
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