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{\bf  \centerline{ Math 262, Spring 2001; Final Exam }}

{This is a closed book exam. Calculators may be used. All working must be shown to receive
full points.}
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\noindent{\bf   1.} {\sl (25 points)} Consider the following $2\times 2$ matrices $A,B,C,D$
over the real numbers:
$$ A = 
\pmatrix{{1} & {2} \cr {-2} &  {1} },\quad
 B = 
\pmatrix{{1} & {2} \cr {2} &  {1} },\quad
 C = 
\pmatrix{{1} & {2} \cr {3} &  {4} },\quad
D = 
\pmatrix{{1} & {2} \cr {0} &  {1} }.$$\smallskip
a) Which of the matrices $A$--$D$ are   orthogonally similar over the real numbers to a
diagonal matrix?

b) Which of the matrices $A$--$D$ are unitarily similar over the complex numbers to a diagonal
matrix?

c) Which of the matrices $A$--$D$ are similar  over the complex numbers to a diagonal matrix? 

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{\noindent \bf 2.} {\sl (25 points)} Prove that for any vectors $x$, $y$  in a real inner
product space 
$V$,
$$\langle x,y\rangle={1\over 4}(\,||x+y\,||^2-||x-y\,||^2\,)$$
What does this tell you about a parallelogram in    $\bf  R^2$ for which the two diagonals are
of equal length?

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{\noindent \bf 3.} {\sl (25 points)} Consider the space $M_{n\times n} (\bf  R)$ of all
$n\times n$  real matrices endowed with the inner product $\langle A,B\rangle=$ 
trace of $B^{\ast}A$ (you may assume that this is an inner product). Consider
the subset $W$ of $M_{n\times n} (\bf  R)$ consisting of those 
matrices of zero trace. Show that $W$ is a subspace of dimension 
$n^2-1$ and find its orthogonal
complement.

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{\noindent \bf 4.} {\sl (25 points)} a) A linear operator $T$ on a $7$ dimensional complex
vector space
$V$ satisfies the following conditions: $\hbox{\rm rank} (T+2)=5$, $\hbox{\rm rank} (
(T+2)^2)=3$,
 $\hbox{\rm rank} ((T+2)^3)=2$, $\hbox{\rm rank} (T-3)=6$,
and $\hbox{\rm rank} ((T-3)^2)=5$. Write down the Jordan canonical form of $T$.\smallskip
b) Write down the possible Jordan canonical forms of a linear operator $S$ on $V$ with
characteristic polynomial $-(t-2)^4t^3$ and minimal polynomial $(t-2)^2t^2$.  



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{\noindent \bf 5.}  {\sl (25 points)} A linear operator $N$ is called
nilpotent if
$N^m=0$ for some $m> 0$.

a) Show that a nilpotent operator $N$ on a finite-dimensional space $V$  has a Jordan
canonical form.

b) Show that $\hbox{Id}_V-N$ is invertible if $N$ is nilpotent.\bigskip 






{\noindent \bf 6.} {\sl (25 points)} a) Let $p(t)$ denote the minimal polynomial 
of a linear operator $T$ on a finite-dimensional vector space over a field $F$. If
$q(t)\in F[t]$ is a polynomial with $q(T)=T_0$ (the zero operator), what is the relation
between $p(t)$ and $q(t)$?
\smallskip b) State necessary and sufficient conditions in terms of $p(t)$ for
diagonalizability of $T$.
\smallskip c) Suppose that $A\in \hbox{Mat}_{n\times n}(\bf C)$ satisfies $A^m=\hbox{Id}_n$
for some
$m\geq 1$. Use (a) and (b) to show that $A$ is similar to a diagonal matrix. 
 \smallskip d) Give an example of a matrix $B\in \hbox{Mat}_{2\times 2}({\bf Z}_2)$ satisfying
$B^2=\hbox{Id}_2$ but such that $B$ is not similar to a diagonal matrix.  
  









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