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

\begin{document}

\maketitle

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\vspace{-1.5\baselineskip}
\section*{Homework week 10}
\vspace{0.5\baselineskip}

Due by the beginning of class on Friday 6th 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 $U$, $V$ be finite-dimensional inner product spaces, and $\phi \in \cL(U,V)$ a linear map.  Moreover, let $u_1,\dots,u_n$ be an orthonormal basis for $U$, $v_1,\dots,v_m$ an orthonormal basis for $V$, and $\sigma_1,\dots,\sigma_n \ge 0$ be the singular values of $\phi$ (where $n=\dim U$, $m = \dim V$), such that
    \[
      \phi(u_i) = \begin{cases} \sigma_i v_i & i \le \min(n,m) \\ 0 & \text{otherwise}. \end{cases}
    \]
    Now let $\Phi \colon U \oplus V \to U \oplus V$ be the linear map $\Phi(u,v) = \big(\phi^\ast(v), \phi(u)\big)$.
    \begin{enumerate}[label={\bf(\alph*)}, leftmargin=0em, labelsep=0.5em, itemsep=0.3\baselineskip, parsep=0.3\baselineskip, listparindent=0em]
      \item Assuming $U \oplus V$ is given the structure of an inner product space, with $\langle (u,v),\, (u',v') \rangle = \langle u, u' \rangle_U + \langle v, v' \rangle_V$, prove that $\Phi$ is self-adjoint.
      \item Compute the eigenvalues, and a corresponding orthonormal basis of eigenvectors, of $\Phi$, in terms of the data $\sigma_i$, $u_i$, $v_i$.
    \end{enumerate}

    [This is a useful trick for using facts about eigenvalues to deduce ones about singular values.]

  \item Let $U=\CC^n$, $V=\CC^m$ with the usual dot product, and $\phi \in \cL(U,V)$ a linear map with matrix $A$ with respect to the standard bases.  Let $k \ge 1$ be an integer.

    Prove that the Schatten $2k$-norm of $\phi$ is given by
    \[
      \|\phi\|_{S( 2k)}^{2k} = \tr\big((\phi^\ast \circ \phi)^k\big) = \sum_{i_1,\dots,i_k =1}^m \sum_{j_1,\dots,j_k =1}^n \prod_{r=1}^k A_{i_r j_r} \overline{A_{i_{r+1} j_r}}
    \]
    where we adopt the convention that $i_{k+1} = i_1$.

  \item Let $U$, $V$ be finite-dimensional inner product spaces, and $\phi \colon U \to V$ a linear map with singular values $\sigma_1 \ge \dots \ge \sigma_n \ge 0$, where $n = \dim U$.
    Let $k$, $1 \le k \le n$, be an integer.

    \begin{enumerate}[label={\bf(\alph*)}, leftmargin=0em, labelsep=0.5em, itemsep=0.3\baselineskip, parsep=0.3\baselineskip, listparindent=0em]
      \item Prove that 
        \[
          \sup_{\substack{\psi \in \cL(U,V) \\ \|\psi\|_{\op} \le 1 \\ \rank(\psi) \le k}} |\langle \phi, \psi \rangle_{\Frob}| = \sum_{i=1}^k \sigma_i.
        \]
        You may use without proof the following fact: if $a_1 \ge \dots \ge a_n \ge 0$ are real numbers and $b_1,\dots,b_n \ge 0$ satisfy $b_i \le 1$ and $\sum_{i=1}^n b_i = k$, then
        \[
          \sum_{i=1}^n a_i b_i \le \sum_{i=1}^k a_i;
        \]
        i.e., the maximum value of the left-hand side under these hypotheses is when $b_1=b_2=\dots=b_k=1$ and $b_{k+1}=\dots=b_n = 0$.

        [{\bf Hint}: the proof of H\"older's inequality for Schatten norms may be useful here.]

      \item Deduce that
        \[
          \|\phi\| = \sum_{i=1}^k \sigma_i
        \]
        defines a norm on $\cL(U,V)$ (called the $k$-th \emph{Ky--Fan norm}).
    \end{enumerate}

\end{enumerate}
\end{document}
