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%{\bf \centerline
%{Math 261 -- Linear Algebra}} 
%{\bf \centerline{Fall 2000 (Quizz 1)}}
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\medskip
{\bf\centerline{ Math 262, Exam 2}}

{Closed book, take home exam. Calculators may be used.}
\bigskip
\noindent{\bf 1} {\sl (25 points)} Consider the following $2\times 2$ matrix $A$ over the
complex numbers
$$ A = 
\pmatrix{a & {{1}\over{2}} \cr b & - {\sqrt{3}\over{2}} }.$$ \smallskip
\noindent (a) Write down the adjoint $A^*$ of $A$.
\smallskip
\noindent (b) For which complex numbers $a$ and $b$ is $A$ unitary?
\smallskip
\noindent (c) 
For which complex numbers $a$ and $b$ is $A$ self-adjoint?
\smallskip
\noindent (d) For which complex numbers $a$ and $b$ is $A$ normal?
\bigskip
\noindent{\bf 2.} {\sl (25 points)} 
\smallskip \noindent  (a) State the spectral theorem for  normal matrices over $\bf C$. 
\noindent \smallskip \noindent(b) Determine the complex eigenvalues and corresponding
eigenspaces in ${\bf C}^2$ of
  $$ A = 
\pmatrix{\cos \theta & {-\sin \theta} \cr \sin \theta &  {\cos \theta} }\in \hbox
{Mat}_{2\times 2}({\bf C})$$ where
$\theta\in
\bf R$ is not a multiple of $\pi$. 
\smallskip\noindent(c)  Compute the matrices (with respect to the standard basis of ${\bf C}^2$) of
the orthogonal projections on the eigenspaces of $A$ (with respect to the standard dot product
on
${\bf C}^2$). Verify the spectral theorem for
$A$.
\smallskip \noindent (d) Is there a real orthogonal matrix $Q$ such that $Q^* A Q$ is
diagonal?

\bigskip 
\noindent{\bf 3.} {\sl (25 points)} Let $T\colon V\rightarrow V$ be a linear operator on a
finite-dimensional complex inner product space $V$.
\smallskip\noindent (a) Show that if $T$ is self-adjoint, then $\langle Tx, x\rangle$ is a
real number for all $x\in V$.
\smallskip \noindent (b) Show that if $\langle Tx, x\rangle=0$ for all $x\in V$, then $T(y)=0$
for all
$y\in V$ (Hint: use that $\langle T(y+z), (y+z)\rangle=0$ and  
$\langle T(y+iz), (y+iz)\rangle=0$ for any $y,z\in V$ where $i=\sqrt{-1}$)  
\smallskip \noindent (c) Show that if  $\langle Tx, x\rangle$ is real for all $x\in V$, then
$T$ is self-adjoint.\bigskip 
\noindent{\bf 4.} {\sl(25 points)}
Let $O(n)$ denote the set of $n \times n$ orthogonal matrices (with 
real entries).  

\smallskip

\noindent(a) Show that every matrix in $O(n)$ has determinant equal to $1$ or $-1$.
\smallskip
\noindent(b) {$SO(n)$ is defined to be the subset of $O(n)$ of all orthogonal matrices 
with determinant one.  Matrices in $SO(n)$ are called ``special 
orthogonal'' matrices.}
Describe the sets $SO(1)$ and $SO(2)$.  (In other words, tell me all 
of the matrices contained in each of these sets.)
\smallskip
\noindent(c)
Suppose that $A \in SO(3)$.  Assume that $a_{13} \neq 0$.  Show that
the values of $a_{11}$, $a_{12}$, $a_{21}$, and $a_{22}$, together
with the sign of $a_{13}$, determine the rest of the entries of $A$.
(If you have this much information about a special orthogonal matrix,
explain how to find all of the entries.)
\smallskip


\noindent{\bf Bonus Question} The set of $n \times n$ ``special unitary'' 
matrices, $SU(n)$, is the set of all unitary matrices (complex 
entries) with determinant one.  Find the simplest description you can of the elements of
$SU(2)$.












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