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\Large Math 554\hfill Autumn 2014

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Homework Set 4\hfill Due Friday October 24
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\kern-.1in{\bf Reading} & Horn \& Johnson, Chapter 1
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\begin{enumerate}
\item
\begin{enumerate}
\item Find the best constants $m,M$ so that $m\|x\|_1\le\|x\|_2\le M\|x\|_1$
for $x\in\C^n$.
\item Find the best constants $m,M$ so that $m\| |A|\|_1\le \| |A|\|_2\le
M\| |A|\|_1$ for $A\in\C^{n\times n}$. 
\item For $A\in\C^{n\times n}$, find the best constants $m,M$ so that $m\|
|A|\|_1\le \|A\|_F\le M\| |A|\|_1$, where
$\|A\|_F=\left(\sum^n_{i,j=1}|a_{ij}|^2\right)^{\frac{1}{2}}$ is the 
Frobenius (Hilbert-Schmidt) norm.
\end{enumerate}
\item It is shown in the notes that a linear functional $f\in\F^{n^\ast}\bs\{0\}$ is
in the closed unit ball of the dual space $\F^\ast_n$ with respect to the
dual norm of a norm $\|\cdot\|$ on $\F^n$ if and only if the real hyperplane
$\{x:\Re f(x)=1\}$ corresponding to $f$ lies completely outside the open
unit ball $\{x:\|x\|<1\}$ of $\|\cdot\|$. Use this criterion to give a proof
that the dual norm to the $\ell^\infty$-norm on $\R^2$ is the $\ell^1$-norm
by explicitly identifying the dual unit ball.
\item Let $(V,\|\cdot\|)$ be a Banach space.
\begin{enumerate}
\item Show that if $L\in\B(V)$ and the operator norm $\|L\|$ satisfies
$\|L\|<1$, then $I-L$ is invertible and
$\frac{1}{1+\|L\|}\le\|(I-L)^{-1}\|\le \frac{1}{1-\|L\|}$.
\item An operator $L\in\B(V)$ is called \emph{invertible} if $L$ is
bijective and $L^{-1}\in\B(V)$. Show that if $L$ is invertible, then the
open ball in $\B(V)$ about $L$ of radius $\frac{1}{\|L^{-1}\|}$ consists
entirely of invertible operators. (So the set of invertible operators in
$\B(V)$ is \emph{open}.)
\end{enumerate}
\item Let $\|\cdot\|$ be a norm on $\R^n$. Identify $\R^n$ with its dual
using the usual Euclidean inner product, so the dual norm is $\|x\|^\ast=
\sup_{\|y\|\le 1}|\lan x,y\ran|$ for $x\in\R^n$. Show that
$\left(\sup_{\|x\|=1}\|x\|_2\right)^2=\sup_{\|x\|=1}\|x\|^\ast$ and
$\left(\inf_{\|x\|=1}\|x\|_2\right)^2=\inf_{\|x\|=1}\|x\|^\ast$. Conclude
that $\|\cdot\|=\|\cdot\|^\ast\Leftrightarrow\|\cdot\|=\|\cdot\|_2$.
\item If $A\in\C^{n\times n}$, recall that $\l$ is an eigenvalue for $A$ if
there is a nonzero $x\in\C^n$ with $Ax=\l x$.
\begin{enumerate}
\item If $\|\cdot\|$ is any submultiplicative norm on $\C^{n\times n}$ and
$\l$ is an eigenvalue of $A$, show that $\|\l\|\le\|A\|$.
\item If $p(z)=z^n+a_{n-1}z^{n-1}+\cdots+a_1z+a_0$ is a monic polynomial
with coefficients $a_i\in\C$, define the \emph{companion matrix} of $p$ to be
$$
C(p)=\left[\begin{array}{cccc}0&1&&\bigcirc\\
&\ddots & \ddots \\
\bigcirc \\
&& 0 & 1\\
-a_0&-a_1&\cdots&-a_{n-1}\end{array}\right]\ .
$$ 
Show that the
eigenvalues of $C(p)$ are precisely the zeroes of $p$.
\item Take $\|\cdot\|=\| |\cdot|\|_1$ and $A=C(p)$ in part (a) to deduce
\emph{Cauchy's bound}: all roots
$z$ of $p$ satisfy $|z|\le 1+\max\{|a_0|,\ldots,|a_{n-1}|\}$.
\item Use $\|\cdot\|=\| |\cdot|\|_\infty$ and deduce \emph{Montel's bound}:
$|z|\le\max\{1,|a_0|+|a_1|+\cdots+|a_{n-1}|\}$ for roots $z$ of $p$.
\item Apply (d) to the polynomial $(z-1)p(z)$ to deduce another bound of
Montel: $|z|\le|a_0|+|a_1-a_0|+\cdots+|a_{n-1}-1|$.
\item Use (3) to prove \index{Kakeya's
Theorem}\index{Theorem!Kakeya's}\emph{Kakeya's Theorem}: if
$f(z)=a_nz^n+a_{n-1}z^{n-1}+\cdots+a_1z+a_0$ is a polynomial with real
nonnegative coefficients satisfying $a_n\ge a_{n-1}\ge \cdots\ge a_1\ge
a_0\ge 0$, then all roots of $f$ lie in the closed unit disk in $\C$.
\end{enumerate}
\end{enumerate}
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