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\begin{document}
\newpage{}
\section*{Lior Silberman's Math 412: Problem set 10, due 5/4/2023}
\begin{lyxlist}{10.}
\item [{P1.}] Recall that a \emph{projection }is a linear map\emph{ }$E$
such that \emph{$E^{2}=E$}. For each $n$ construct a projection
$E_{n}\colon\R^{2}\to\R^{2}$ of norm at least $n$ ($\R^{n}$ is
equipped with the Euclidean norm unless specified otherwise). Prove
for yourself that the norm of an \emph{orthogonal} projection is $1$.
\end{lyxlist}
\begin{center}
\textbf{Difference and differential equations}\bigskip{}
\par\end{center}
\begin{lyxlist}{10.}
\item [{P2.}] Let $A=\begin{pmatrix}0 & 1\\
1 & 1
\end{pmatrix}$. Let $\vv_{0}=\begin{pmatrix}0\\
1
\end{pmatrix}$.
\begin{lyxlist}{10.}
\item [{(a)}] Find $S$ invertible and $D$ diagonal such that $A=S^{-1}DS$.
\item [{--}] Prove for yourself the formula $A^{k}=S^{-1}D^{k}S$.
\item [{(b)}] Find a formula for $\vv_{k}=A^{k}\vv_{0}$, and show that
$\frac{\vv_{k}}{\norm{\vv_{k}}}$ converges for any norm on $\R^{2}$.
\item [{RMK}] You have found a formula for Fibbonacci numbers (why?), and
have shown that the real number $\frac{1}{2}\left(\frac{1+\sqrt{5}}{2}\right)^{n}$
is exponentially close to being an integer.
\item [{RMK}] This idea can solve any \emph{difference equation}, and also
\emph{differential equations}.\bigskip{}
\end{lyxlist}
\item [{1.}] We will analyze the differential equation $u''=-u$ with initial
data $u(0)=u_{0}$, $u'(0)=u_{1}$.
\begin{lyxlist}{10.}
\item [{(a)}] Let $\vv(t)=\begin{pmatrix}u(t)\\
u'(t)
\end{pmatrix}$. Show that $u$ is a solution to the equation iff $\vv$ solves
\[
\vv'(t)=\begin{pmatrix}0 & 1\\
-1 & 0
\end{pmatrix}\vv(t)\,.
\]
\item [{(b)}] Let $W=\begin{pmatrix}0 & 1\\
-1 & 0
\end{pmatrix}$. Find a formula for $W^{n}$ and express $\exp(Wt)=\sum_{k=0}^{\infty}\frac{W^{k}t^{k}}{k!}$
as a matrix whose entries are standard power series.
\item [{(c)}] Sum the series and show that $u(t)=u_{0}\cos(t)+u_{1}\sin(t)$.
\item [{(d)}] Find a matrix $S$ such that $W=S\begin{pmatrix}i & 0\\
0 & -i
\end{pmatrix}S^{-1}$. Evaluate $\exp(Wt)$ again, this time using $\exp(Wt)=S\left(\exp\begin{pmatrix}it & 0\\
0 & -it
\end{pmatrix}\right)S^{-1}$.\bigskip{}
\end{lyxlist}
\item [{DEF}] The \emph{companion matrix }associated to the polynomial
$p(x)=x^{n}-\sum_{i=0}^{n-1}a_{i}x^{i}$ is
\[
C=\begin{pmatrix}0 & 1 & 0 & 0 & 0\\
0 & 0 & 1 & 0 & 0\\
0 & 0 & 0 & \ddots & 0\\
0 & 0 & 0 & 0 & 1\\
a_{0} & a_{1} & \cdots & a_{n-2} & a_{n-1}
\end{pmatrix}\,.
\]
\item [{2.}] A sequence $\left\{ x_{k}\right\} _{k=0}^{\infty}$ is said
to satisfy a \emph{linear recurrence relation} (or a \emph{difference
equation})\emph{ }if for each $k$,
\[
x_{k+n}=\sum_{i=0}^{n-1}a_{i}x_{k+i}\,.
\]
\begin{lyxlist}{10.}
\item [{(a)}] Define vectors $\vv^{(k)}=(x_{k-n+1},x_{k-n+2},\ldots,x_{k})$.
Show that $\vv^{(k+1)}=C\vv^{(k)}$ where $C$ is the companion matrix
above.\bigskip{}
\item [{(b)}] Find $x_{100}$ if $x_{0}=1$, $x_{1}=2$, $x_{2}=3$ and
$x_{n}=x_{n-1}+x_{n-2}-x_{n-3}$.\\
\emph{hint:} Find the Jordan canonical form of $\begin{pmatrix} & 1\\
& & 1\\
-1 & 1 & 1
\end{pmatrix}$.\bigskip{}
\end{lyxlist}
\end{lyxlist}
\newpage{}
\begin{lyxlist}{10.}
\item [{3.}] Let $C$ be the companion matrix associated with the polynomial
$p(x)=x^{n}-\sum_{k=0}^{n-1}a_{k}x^{k}$.
\begin{lyxlist}{10.}
\item [{(a)}] Show that $p(x)$ is the characteristic polynomial of $C$.
\item [{(b)}] Show that $p(x)$ is also the minimal polynomial.
\item [{--}] For parts (c),(d) fix a non-zero root $\lambda$ of $p(x)$.
\item [{(c)}] Find (with proof) an eigenvector with eigenvalue $\lambda$.
\item [{({*}{*}d)}] Let $g$ be a polynomial, and let $\vv$ be the vector
with entries $v_{k}=\lambda^{k}g(k)$ for $0\leq k\leq n-1$. Show
that, if the degree of $g$ is small enough (depending on $p,\lambda$),
then $\left(\left(C-\lambda\right)\vv\right)_{k}=\lambda\left(g(k+1)-g(k)\right)\lambda^{k}$
and (the hard part) that
\[
\left((C-\lambda)\vv\right)_{n-1}=\lambda\left(g(n)-g(n-1)\right)\lambda^{n-1}\,.
\]
\item [{({*}{*}e)}] Find the Jordan canonical form of $C$.\bigskip{}
\end{lyxlist}
\item [{4.}] Consider now \emph{differential} equation $\frac{\diff}{\dft}\vv=B\vv$
where $B$ is at in PS8 problem 1.
\begin{lyxlist}{10.}
\item [{(a)}] Find matrices $S,D$ so that $D$ is in Jordan form, and
such that $B=SDS^{-1}$.
\item [{(b)}] Find $\exp(tD)$ as in 1(b) by computing a formula for $D^{n}$
and summing the series.
\item [{(c)}] Find the solution such that $\vv(0)=\begin{pmatrix}0 & 1 & 1 & 0\end{pmatrix}^{\textrm{t}}$.\bigskip{}
\end{lyxlist}
\end{lyxlist}
\begin{center}
\textbf{Power series}\bigskip{}
\par\end{center}
\begin{lyxlist}{10.}
\item [{5.}] Let $A=\begin{pmatrix}z & 1\\
0 & z
\end{pmatrix}$ with $z\in\C$.
\begin{lyxlist}{10.}
\item [{(a)}] Find a simple formula for the entries of $A^{n}$.
\item [{(b)}] Use your formula to decide the set of $z$ for which $\sum_{n=0}^{\infty}A^{n}$
converge, and give a formula for the sum.
\item [{(c)}] Show that the sum is $\left(\Id-A\right)^{-1}$ when the
series converges.\bigskip{}
\end{lyxlist}
\item [{6.}] For any matrix $A\in M_{n}(\C)$ show that $\sum_{n=0}^{\infty}z^{n}A^{n}$
converges for $\left|z\right|<\frac{1}{\rho(A)}$.\\
\emph{Hint}: see PS9 problem 3.
\end{lyxlist}
\begin{center}
\textbf{Supplementary problems}\bigskip{}
\par\end{center}
\begin{lyxlist}{10.}
\item [{A.}] Consider the map $\Tr\colon M_{n}(F)\to F$.
\begin{lyxlist}{10.}
\item [{(a)}] Show that this is a continuous map.
\item [{(b)}] Find the norm of this map when $M_{n}(F)$ is equipped with
the $L^{1}\to L^{1}$ operator norm (see PS9 Problem 2(a)).
\item [{(c)}] Find the norm of this map when $M_{n}(F)$ is equipped with
the Hilbert--Schmidt norm (see PS9 Problem A).
\item [{({*}d)}] Find the norm of this map when $M_{n}(F)$ is equipped
with the $L^{p}\to L^{p}$ operator norm. Find the matrices $A$ with
operator norm $1$ and trace maximal in absolute value.\bigskip{}
\end{lyxlist}
\item [{B.}] Call $T\in\End_{F}(V)$ \emph{bounded below} if there is $K>0$
such that $\norm{T\vv}\geq K\norm{\vv}$ for all $\vv\in V$.
\begin{lyxlist}{10.}
\item [{(a)}] Let $T$ be boudned below. Show that $T$ is invertible,
and that $T^{-1}$ is a bounded operator.
\item [{({*}b)}] Suppose that $V$ is finite-dimensional. Show that every
invertible map is bounded below.\bigskip{}
\end{lyxlist}
\item [{C.}] (The supremum norm and the Weierestrass $M$-test) Let $V$
be a complete normed space.
\begin{lyxlist}{10.}
\item [{DEF}] For a set $X$ call $f\colon X\to V$ \emph{bounded }if there
is $M>0$ such that $\norm{f(x)}_{V}\leq M$ for all $x\in X$ in
which case we write $\norm{f}_{\infty}=\sup_{x\in X}\norm{f(x)}_{V}$
(equivalently, $f$ is bounded if $x\mapsto\norm{f(x)}_{V}$ is in
$\ell^{\infty}(X)$).
\item [{(a)}] Show that $\ell^{\infty}(X;V)$ is a vector space (this doesn't
use completeness of $V$).
\item [{(b)}] Show that $\ell^{\infty}(X;V)$ is complete.
\item [{DEF}] Now suppose that $X$ is a metric space (or, more generally,
a topological space). Let $C(X;V)$ denote the space of \emph{continuous}
functions $X\to V$ and let $\Cb(X;V)=C(X;V)\cap\ell^{\infty}(X;V)$
be the space of \emph{bounded} continuous functions, the latter equipped
with the $\ell^{\infty}$-norm.
\item [{(c)}] Show that $\Cb(X;V)$ is a closed subspace of $\ell^{\infty}(X;V)$.
Conclude that it is complete.
\item [{COR}] Deduce Weirestrass's $M$-test: $f_{n}\colon X\to V$ are
continuous and satisfy $\norm{f_{n}}_{\infty}\leq M_{n}$ with $\sum_{n}M_{n}<\infty$
then $\sum_{n}f_{n}$ converges to a continuous function of norm at
most by $\sum_{n}M_{n}$.
\end{lyxlist}
\end{lyxlist}
\end{document}