% LaTeX file -- M245, Fall 2012, Assignment 4

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\begin{center}
\textbf{\large Math 245 \\ Assignment 4\\
Due  Wednesday October 31}
\end{center}
\vskip 2em 


\begin{Ex}
\item  Let $T \in \L(V)$ where $V$ is a vector space over an algebraically closed field.
\begin{parts}
\item Show that $\A'(T):= \{A \in \L(V) : AT=TA\}$ is an algebra containing $\A(T)$,
and every $A \in\A'(T)$ commutes with every $S \in \A(T)$.
\item Compute $\A'(J_n)$.
\item Show that if $T$ has a cyclic vector, then $\A'(T)=\A(T)$.\\
\hint any $A \in\A'(T)$ commutes with the projections $E_i$ onto $\ker((T-\lambda_i I)^{d_i})$.
\item Show that if $\A'(T)=\A(T)$, then $T$ has a cyclic vector.
\end{parts}


\item Consider the linear recursion $x_{k+4}-8x_{k+3}+24x_{k+2}-32x_{k+1}+16x_k = 0$.
\begin{parts}
\item Set this up as a linear algebra problem as in class. Find the Jordan form for the matrix $T$.
\item Find an explicit basis that puts it in Jordan form.
\item Find a formula for $x_k$ given $(x_0,x_1,x_2,x_3)=(1,2,3,4)$.
\end{parts}


\item Consider a (fictional) chemical in solution in a 3-chambered
bottle. Because of the different conditions in the three chambers,
molecules of the chemical pass across the membrane from one chamber to
another with different probabilities. In one minute, the probability \vspace{.3ex}
that a molecule in chamber $C_j$ moves to chamber $C_i$ is $p_{ij}$ where
$
A = \big[ p_{ij} \big] = \begin{bmatrix}.4&.2&0\\.3&.5&.2\\.3&.3&.8 \end{bmatrix}$.
Find the limit state.


\item \begin{parts}
\item Suppose that $a_k$ are  real numbers such that $\displaystyle \sum_{k=1}^n a_k = 1$. 
Prove that $\displaystyle \sum_{k=1}^n 2^ka_k^2 > 1.$
\item Find the minimum of $\displaystyle \sum_{k=1}^n 2^ka_k^2$ as $\{ a_k \}$ runs over all real solutions
of $\displaystyle \sum_{k=1}^n a_k = 1$.
\end{parts}


\item 
\begin{parts}
\item If $V$ is a complex inner product space, show that
\[ \ip{x,y} = \tfrac14\big( \|x+y\|^2 - \|x-y\|^2 +i\|x+iy\|^2 -i\|x-iy\|^2  \big). \]
\item If $V$ is a real inner product space, show that if $\|x+y\|^2 = \|x\|^2 + \|y\|^2$,
then $x$ and $y$ are orthogonal. What can you deduce from this identity in a complex inner product space?
\end{parts}


\item \textbf{Bonus.} Let $A= \big[ a_{ij} \big]$ be an $n \times m$ matrix with complex entries.
Define 
\[ R = \max_{1\le i \le n} \sum_{j=1}^m |a_{ij}|  \qand C = \max_{1\le j \le m} \sum_{i=1}^n |a_{ij}| .\]
For $x=(x_1,\dots,x_m)^t \in \bC^m$ and $y=(y_1,\dots,y_n)^t \in \bC^n$, prove that  
\[ |\ip{Ax,y}| \le \sqrt{RC}\, \|x\|_2\,\|y\|_2 .\]


\end{Ex}
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