\documentclass[10pt,reqno,letterpaper]{amsart}

\usepackage{tabularx}
\usepackage{enumitem}
\usepackage{url}

\newcommand{\cL}[0]{\mathcal{L}}
\newcommand{\cC}[0]{\mathcal{C}}
\newcommand{\RR}[0]{\mathbb{R}}
\newcommand{\NN}[0]{\mathbb{N}}
\newcommand{\FF}[0]{\mathbb{F}}
\newcommand{\CC}[0]{\mathbb{C}}
\newcommand{\ZZ}[0]{\mathbb{Z}}
\newcommand{\spn}[0]{\operatorname{span}}

\pdfpagewidth 8.3in
\pdfpageheight 11.7in
\setlength\textwidth{6.15in}
\setlength\textheight{9.3in}
\setlength\topmargin{-0.1in}
\setlength\oddsidemargin{0.1in}
\setlength\evensidemargin{0.1in}


%\pagenumbering{gobble}
\title{MATH 202A \\ Applied Algebra I \\ Fall 2019}

\begin{document}

\maketitle

\def\id{\operatorname{id}}
\def\op{\operatorname{op}}
\def\im{\operatorname{im}}
\def\ker{\operatorname{ker}}
\def\tr{\operatorname{tr}}
\def\eps{\varepsilon}
\def\spec{\operatorname{spec}}
\def\Frob{\operatorname{Frob}}

\vspace{-1.5\baselineskip}
\section*{Homework week 9}
\vspace{0.5\baselineskip}

Due by the beginning of class on Monday 2nd December (hand in via Gradescope).
\vspace{0.5\baselineskip}

\begin{enumerate}[label={\bf\arabic*.}, itemsep=0.8\baselineskip, leftmargin=2em, labelsep=2em, listparindent=0em, parsep=0.3\baselineskip]
  \item Let $V$, $W$ be finite-dimensional vector spaces.
      Prove that the map
        \[
          \begin{aligned}
            \cL(V,W) &\to \cL(W,V)^\ast \\
            \phi &\mapsto \big( \psi \mapsto \tr(\phi \circ \psi) = \tr(\psi \circ \phi) \big)
          \end{aligned}
        \]
        is an isomorphism.

        [So, a matrix norm on $\cL(V,W)$ has a corresponding dual norm on $\cL(W,V)$, etc..]

  \item Consider $V = \CC^n$, $W = \CC^m$.  For a linear map $\phi \colon V \to W$ represented my an $m \times n$ matrix $A$, prove that:---
    \begin{enumerate}[label={\bf(\alph*)}, leftmargin=0em, labelsep=0.5em, itemsep=0.3\baselineskip, parsep=0.3\baselineskip, listparindent=0em]
      \item $\|\phi\|_{\ell^1 \to \ell^1} = \max_{j=1}^n \sum_{i=1}^m |A_{ij}|$;
      \item $\|\phi\|_{\ell^\infty \to \ell^\infty} = \max_{i=1}^m \sum_{j=1}^n |A_{ij}|$;
      \item $\|\phi\|_{\ell^1 \to \ell^\infty} = \|A\|_{v,\infty}$;
      \item $\|\phi\|_{\ell^\infty \to \ell^1} = \|A\|_{v,1}$.
    \end{enumerate}

  \item Let $U$, $V$, $W$ be inner product spaces, and let $\phi \in \cL(U,V)$.  Define
    \[
      \begin{aligned}
        \Phi \colon \cL(V,W) &\to \cL(U,W) \\
        \psi &\mapsto \psi \circ \phi.
      \end{aligned}
    \]
    Prove that the operator norm $\|\Phi\|_{\Frob(\cL(V,W)) \to \Frob(\cL(U,W))}$ with respect to the Frobenius norms on $\cL(V,W)$ and $\cL(U,W)$, is exactly the operator norm $\|\phi\|_{U \to V}$.

  \item
    Let $V$ be a finite-dimensional inner product space.  
    \begin{enumerate}[label={\bf(\alph*)}, leftmargin=0em, labelsep=0.5em, itemsep=0.3\baselineskip, parsep=0.3\baselineskip, listparindent=0em]
      \item Prove that if $\phi,\psi \in \cL(V,V)$ are both self-adjoint then $\langle \phi, \psi \rangle_{\Frob}$ is real.

        \vspace\baselineskip
      \noindent
      Given $v \in V$, let $\theta_v \in \cL(V,V)$ denote the linear map $u \mapsto \langle u, v \rangle\, v$.  Consider the subset $\cC \subseteq \cL(V,V)$ given by
      \[
        \cC = \left\{ \sum_{i=1}^m a_i \theta_{v_i} \colon m \ge 0,\ v_1,\dots,v_m \in V,\ a_1,\dots,a_m \ge 0 \right\};
      \]
      i.e., the set of non-negative sums of elements $\theta_v$.
    
      \item Let $\phi \in \cL(V,V)$ be self-adjoint.  Prove that the following are equivalent:---
      \begin{enumerate}[label={\bf(\roman*)}, leftmargin=3em, labelsep=0.5em, itemsep=0.3\baselineskip, parsep=0.3\baselineskip, listparindent=0em]
        \item $\phi$ is non-negative definite;
        \item $\phi \in \cC$;
        \item for all $\psi \in \cC$ we have $\langle \phi, \psi \rangle_{\Frob} \ge 0$.
      \end{enumerate}
      [If you wish, you can say that ``the non-negative definite cone $\cC$ is self-dual''.]
    \end{enumerate}

\end{enumerate}
\end{document}
