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\fancyhead[L]{\Large \textbf{Math 207 Fall 2014}}
\fancyhead[R]{\large \textbf{Problem Set 2}}


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\bp In quantum mechanics one considers the three matrices $\sigma^x,
\sigma^y,$ and $\sigma^z$.  
$$
\sigma^x=\begin{bmatrix}
0 & 1 \\
1 & 0 
\end{bmatrix}
\qquad 
\sigma^y=\begin{bmatrix}
0 & -i \\
i & 0 
\end{bmatrix}
\qquad 
\sigma^z=\begin{bmatrix}
1 & 0 \\
0 & -1 
\end{bmatrix}
$$
In quantum computing theory, these
matrices represent fundamental operations on a single quantum bit of quantum
information.  For example, $\sigma^x$ is the ``Quantum NOT'' operation.
\ben
\item Compute the eight products: $\sigma^i \sigma^j$ for $i,j=x,y,z.$
\item Find the eigenvalues of $\sigma^x,
\sigma^y,$ and $\sigma^z$.
\een
\ep


\bp Consider the inner product space $(V,\<\,,\,\>)$ where $V$ is the
space of all infinitely differentiable functions $f:\R\to \R$
satisfying $f(x)=f(x+1)$ for all $x\in \R$
and $\<f,g\>=\int_0^1 f(x)g(x)dx$.
Let $\sigma:V \to V$ defined by $\sigma(f)=f'$.
\ben
\item Prove that $\sigma$ is a skew-symmetric linear operator.
\item Prove that $\sigma^2$ is a negative semi-definite symmetric operator.
\een
Recall:  
A linear operator $\tau$ on
an inner product space $V$ is 
negative semi-definite if and only if $\<\tau(v),v\>\leq 0$ for all
$v\in V$.  
\ep



\section*{The Matrix Exponential}

\begin{definition} For any $n\times n$ complex matrix $A$, define the
  matrix $e^A$ by
\begin{equation}\label{exp}
e^A=I+A+\frac{A^2}{2!}+\frac{A^3}{3!}+\cdots
\end{equation}
One can prove that the series defined above converges absolutely for all matrices $A$.
\end{definition}



\bp Compute $e^A$ and $e^B$ for $$
A=\begin{bmatrix}1&0\\1&1\end{bmatrix}\text{ and }
B=\begin{bmatrix}3i&0&-2+i\\3-2i&-2&1+2i\\0&0&6\end{bmatrix}.$$
\ep


\bp Prove that if $A$ is similar to $B$ then $e^A$ is similar to
$e^B$.
\ep

% \bp $e^{A+B}$ versus $e^A e^B$
% \ben 
% \item Find two matrices $A$ and $B$ for which 
% $e^{A+B}\neq e^A e^B.$
% \item Suppose that $A$ and $B$ are $n\times n$ commuting matrices.
% Show that $e^{A+B}=e^Ae^B.$  
% \een
% \ep

\bp Prove that $e^{\operatorname{trace}(A)}=\det(e^A)$.
\ep


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