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\title{MATH 351, Fall 2017, Homework \#9}

\author{\underline{\bf Due Friday, November 10, 2017}}



\maketitle

\begin{problem}[cf. 6.120, p. 394] Let $A$ be an $n\times n$ real matrix such that $A = A^t$. We will say that $A$ is \underline{positive semidefinite} if $(A\vec x)\cdot \vec x\geq 0$ for all $\vec x\in \mathbb R^n$. 
\end{problem}

Show that if $A = B^t B$ for some real $k\times n$ matrix $B$, then $A$ is positive semidefinite.



\begin{problem} Show the following properties are equivalent to $A= A^t$ being positive semidefinite:
\end{problem}

\begin{itemize}
\item[a)] All eigenvalues of $A$ are non-negative.
\item[b)] There is an $n\times n$ real matrix $C$ such that $C = C^t$ and $A = C^2$.
\end{itemize}

\begin{problem}[cf. 6.118, p. 394] Find a $2\times 2$ real matrix, equal to its transpose, with positive entries but which is not positive semidefinite. 
\end{problem}



\begin{problem} Let $P$ be an $n\times n$ real matrix such that $P^t = P$ and $P^2 = P$. Show that $P$ is the matrix of the orthogonal projection onto the subspace of all vectors $\vec x\in \mathbb R^n$ such that $P\vec x = \vec x$.
\end{problem}





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