\documentclass{amsart}

\usepackage{amsmath}
\usepackage{amssymb}

\input{macros}

\newcommand{\cH}{\mathcal{H}}
\newcommand{\bB}{\mathbb{B}}
\newcommand{\bH}{\mathbb{H}}
\newcommand{\cS}{\mathcal{S}}
\newcommand{\ip}[2]{\left\langle #1,\, #2\right\rangle}
\newcommand{\norm}[1]{\left\Vert #1\right\Vert}


\begin{document}
\noindent
\Large Math 554\hfill Autumn 2014

\noindent
Homework Set 6\hfill Due Friday, November 7
\bigskip

\begin{flushleft}
\begin{tabular}{lp{5.5in}}
\kern-.1in{\bf Reading} & Horn \& Johnson, Chapter 4 - 5
\end{tabular}
\end{flushleft}


\begin{enumerate}
\item Show that $U\in\C^{n\times n}$ is unitary iff $\exists$ a Hermitian
$H\in\C^{n\times n}$ for which $U=e^{iH}$. 
\item Use the Schur Triangularization Theorem to show that every matrix
$A\in\C^{n\times n}$ is ``almost'' diagonalizable in the following two
senses.
\begin{enumerate}
\item Given $\e>0$, there is a matrix $\widetilde A\in\C^{n\times n}$ with
distinct eigenvalues for which $\|A-\widetilde A\|_F<\e$. 
\item Given $\e>0$, there is an upper triangular $T$ similar to $A$ for
which $|t_{ij}|<\e$ for all $i<j$.
\end{enumerate}
\item Let $A, B\in\C^{n\times n}$ be diagonalizable. Show that $A$ and $B$
are \emph{simultaneously diagonalizable} (i.e., $\exists$ one invertible
$S\in\C^{n\times n}$ for which both $S^{-1}AS$ and $S^{-1}BS$ are diagonal)
iff $AB=BA$, as follows:
\begin{enumerate}
\item Show that if $A,B$ are simultaneously diagonalizable, then $AB=BA$.
\item Suppose $AB=BA$. Let $\l_1,\ldots, \l_k$ be the distinct eigenvalues
of $A$, with eigenspaces $E_1,\ldots, E_k$ and associated projections
$P_1,\ldots, P_k$. Show that $BE_i\subset E_i$ for each $i$ and deduce that
$BP_i=P_iB$ for each $i$.
\item Suppose $AB=BA$. Let $\{v_1,\ldots, v_n\}$ be a basis of $\C^n$
consisting of eigenvectors of $B$. Show that for each $i$, $1\le i\le i$,
the vectors $\{P_iv_1,\ldots, P_iv_n\}$ span the subspace 
$E_i$ (where $E_i,P_i$ are as in
part (b)). Also show that each nonzero $P_iv_j$ is an eigenvector of $B$.
\item Suppose $AB=BA$. Deduce that there is a basis for $\C^n$ consisting of
vectors which are eigenvectors for both $A$ and $B$. Conclude that $A$ and
$B$ are simultaneously diagonalizable.
\end{enumerate}
\item Prove the real version of the Schur Theorem: If $A\in\R^{n\times n}$,
there is an orthogonal matrix $V\in\R^{n\times n}$ so that
$V^TAV=\left[\begin{array}{ccc}
A_1&&\ast\\
&\ddots\\
0&&A_k\end{array}\right]\equiv B$ is block upper triangular, where each
$A_i$ is either a real $1\times 1$ matrix or a real $2\times 2$ matrix with
eigenvalues a complex conjugate pair $\l\ne\bar\l$. (The matrix
$B\in\R^{n\times n}$ is often called
\index{quasi-upper-triangular}\emph{quasi-upper-triangular}.)
\item
\begin{enumerate}
\item What are the possible Jordan canonical forms for $A$ if
$p_A(t)=(t+3)^4(t-4)^2$?
\item Find the Jordan canonical form for $A=\left(\begin{array}{ccc}0&-1&-1\\
1&-2&0\\ 0&0&-2\end{array}\right)$.
\end{enumerate}
%\item Do Problem 1 on p.\ 76 of Horn and Johnson.
\item
Let $\bH$ be a real Hilbert space with inner product $\ip{\cdot}{\cdot}$
and inner product norm $\norm{\cdot}$. 
A set $K\subset \bH$ is said to be a cone if
$\alpha K\subset K$ for all $\alpha>0$. Given a set $C\subset \bH$, define the {\it polar}
of $C$ to be the set $C^\circ:=\set{y\in\bH}{\ip{y}{x}\le 1\ \forall\, x\in C}$.
\begin{enumerate}
\item Show that $K\subset \bH$ is a convex cone if and only if (i) $\alpha K\subset K$ for all 
$\alpha>0$ and (ii) $K+K\subset K$.
\item Let $\bB$ be the closed unit ball in $\bH$. Show that $\bB^\circ=\bB$.
\item Given $C\subset \bH$, show that $C^\circ$ is always a closed convex set.
\item Given $C\subset \bH$, show that $(C^\circ)^\circ$ is the closed convex hull of
$C$, i.e., the intersection of all closed convex sets containing $C$.
\item Given a non-empty closed convex cone $K\subset \bH$ show that
$$K^\circ:=\set{y\in\bH}{\ip{y}{x}\le 0\ \forall\, x\in K}.$$
\item Let $K\subset \bH$ be a non-empty closed convex cone.
Given $z,x,y\in\bH$, show that the following 
statements are equivalent:
\begin{enumerate}
\item $z=x+y$ with $x\in K,\ y\in K^\circ$ and $\ip{x}{y}=0$.
\item $x=P_Kz$ and $y=P_{K^\circ}z$, where $P_K$ and $P_{K^\circ}$ are the projections onto
$K$ and $K^\circ$, respectively.
\end{enumerate}
\end{enumerate}
\end{enumerate}



\end{document}
