% LaTeX file -- M245, Fall 2012, Assignment 5

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\begin{document}
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\textbf{Reminder:} Midterm is Tuesday November 6, 4:30-6:20 in MC 4064.
\vskip6mm

%%%%%%%   TITLE   %%%%%%
\begin{center}
\textbf{\large Math 245 \\ Assignment 5\\
Due  Wednesday November 14}
\end{center}
%\vskip 2em 



\begin{Ex}
\item \begin{parts}
\item Apply the Gram-Schmidt process to 
\[
v_1= \begin{bmatrix}1\\2\\2\\4\end{bmatrix},\qquad
v_2= \begin{bmatrix}1\\1\\1\\0\end{bmatrix},\qquad
v_3= \begin{bmatrix}0\\1\\0\\2\end{bmatrix},\qquad
v_4= = \begin{bmatrix}\phantom{-}4\\\phantom{-}0\\-4\\\phantom{-}1\end{bmatrix}.
\]

\item Find the matrix (with respect to the standard basis) of the
orthogonal projection of $\bR^4$ onto $\spn\{v_1,v_2\} $.
\end{parts}


\item Find an orthonormal basis which diagonalizes  
$ T= \left[\begin{array}{rrr}
0 &20&-3\\20&0&-3\\-3&-3&13
\end{array} \right] ,
$
and find a unitary $U$ so that $U^*TU$ is diagonal.

\item 
Let $V$ be a vector space over $\bC$ with a positive semidefinite sesquilinear form $\ip{\cdot,\cdot}$. i.e.\ $\ip{v,v} \ge 0$. \vspace{-2ex}
\begin{parts}
\item Show that $N := \{v : \ip{v,v}=0 \}$ is a subspace.\\
\hint first show that $N = \{v : \ip{v,u}=0 \FORAL u \in V \}$.
\item Put a relation on $V$ by $u \equiv v$ if and only if $u-v \in N$.
Prove that this is an equivalence relation.
\item Let $W = V/N$ denote the set of equivalence classes. Show that this is a vector space
with the operations $\alpha [v] = [\alpha v]$ and $[u] + [v] = [u+v]$. That is, show that these
operations are well defined, and that $W$ satisfies the properties of a vector space.
\item Define an inner product on $W$ by $\ip{[u],[v]} = \ip{u,v}$.
Prove that this is well defined. Then show that it is a positive definite sesquilinear form on $W$.
\end{parts}


\item 
\begin{parts}
\item  Hence show that $T \in \L(V)$ on a complex inner product space $V$ is \textit{positive}
(i.e. self-adjoint with non-negative eigenvalues) if and only if $\ip{Tx,x}\ge 0$ for all $x\in V$.\\
\hint first show that $T$ is self-adjoint by using this analogue of Assignment 3, 5(a):
\[
 \ip{Tx,y}=\tfrac{1}{4}\big( \ip{T(x\!+\!y),x\!+\!y} \!-\! \ip{T(x\!-\!y),x\!-\!y}
  \!+\! i \ip{T(x\!+\!iy),x\!+\!iy} \!-\! i \ip{T(x\!-\!iy),x\!-\!iy}\big).
\]

\item If $T$ is any matrix in $\L(V)$, show that $T^*T$ is positive.
\item Find a matrix $T$ acting on $\bR^2$ which is not self-adjoint, but
$\ip{Tx,x} = 0$ for all $x\in\bR^2$.
\end{parts}

\item \label{simdiag}
Show that two normal matrices $N$ and $M$ can be diagonalized by the
same orthonormal basis (\textit{simultaneous diagonalization}) if and only if $NM=MN$.\\
\hint $(\Longleftarrow)$ start with one, say $N$, already diagonalized, and compute $NM=MN$ to
find conditions on the matrix form of $M$.

\item \textbf{Bonus.}   
Show that a positive matrix $A$ has a unique positive square root. \\
\hint Diagonalize $A$ and find the `obvious' square root $B$. Notice that there
must be a polynomial $p$ so that $B=p(A)$. Hence show that if $C$ is another
square root, then $BC=CB$. Apply problem \ref{simdiag}.

\end{Ex}
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