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\begin{document}
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\title{Assignment 5, CO666}
\author{
          University of Waterloo\\
          Department of Combinatorics and Optimization\\
          Waterloo, Ontario N2L 3G1, Canada\\
          }
\date{\today}
          \maketitle

\begin{enumerate}
\item
\label{xAx}
Suppose that $A,B \in \Sn$ are two given symmetric $n \times n$
matrices and there exists $x \in \Rn$ such that $x^t A x > 0.$
Prove that the following two statments are equivalent:
\begin{enumerate}
\item
\[ x^t A x \geq  0 \mbox{ implies } x^t B x \geq  0
\]
\item
There exists $\lambda \geq 0, \Lambda \succeq 0$ (positive semidefinite)
such that
\[ 
B = \lambda A + \Lambda.
\]

\end{enumerate}
\item
Suppose that $A,B,C \in \Sn$ are three given symmetric $n \times n$
matrices and there exists $x \in \Rn$ such that 
both $x^t A x > 0$ and $x^t B x > 0.$
Show by example that the statement
\[ x^t A x \geq  0, x^t B x \geq  0 \mbox{ implies } x^t C x \geq  0
\]
does {\bf not} imply the existence of $\lambda, \mu \geq 0,$ such that
$C \succeq \lambda A + \mu B.$

\item
Bertsekas Page 436 \#5.1.6

\end{enumerate}

\end{document}
