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\pagestyle{fancy} \lhead{{\sf Homework 13 $|$ Due April 25
    (Thursday) }} \rhead{\thepage} \cfoot{{\sf MthSc 853 $|$ Linear
    Algebra $|$ Spring 2013 $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent Throughout, all vector spaces are finite-dimensional over a
field $K$. You may assume that these spaces are endowed with a
Euclidean structure, though this is not always needed. The set of
linear maps from $X$ to $U$ is denoted by $\Hom(X,U)$. \\

\noindent\emph{Read}: Lax, Chapter 10, pages 143--153, and Appendix 4,
pages 313--316.

\begin{enumerate}

\item Define the \emph{index} of a real symmetric matrix $A$ to be the
  number of strictly positive eigenvalues minus the number of strictly
  negative eigenvalues. Suppose $A$ and $B$ are real symmetric
  matrices and $x^TAx\leq x^TBx$ for all $x\in X$. Prove that the
  index of $A$ is at most the index of $B$.

  

% Lax Exercise 10.3, 10.4, 10.5
\item Let $A$, $B$ be self-adjoint mappings with $0<A<B$. Here, $<$ is
  the partial ordering of self-adjoint mappings where $A<B$ iff $N-A$
  is positive-definite.
  \begin{enumerate}
  \item Show by example that the \emph{symmetrized product} $S:=AB+BA$
    need not be positive-definite.
  \item Show that $A^{1/k}<B^{1/k}$ if $k$ is a power of $2$.
  \item Show that $\log A\leq\log B$.
  \item Show by example that $A^2<B^2$ need not hold.
  \end{enumerate}



% Lax Exercise A4.2 
\item Let $U$ and $V$ be vector spaces over a field $K$. Define the map 
  \[
  \varphi\colon U\otimes V\longrightarrow\Hom(U',V)\,,\qquad
  \varphi\colon u\otimes v\longmapsto \{\ell\mapsto(\ell,u)v\}\,,
  \]
  and extend linearily to arbitrary elements $\sum(u_i\otimes v_i)$ in
  $U\otimes V$. Prove that $\varphi$ is an isomorphism.

  
  
\item Let $U$, $V$, and $X$ by vector spaces over a field
  $K$. Define a map
  \[
  \iota\colon U\times V\longrightarrow U\otimes V\,,\qquad
  \iota(u,v)=u\otimes v\,.
  \]
  \begin{enumerate}
  \item Prove that $\iota$ is bilinear.
  \item Prove that if $A\in\Hom(U\otimes V,X)$, then
    $\alpha:=A\circ\iota$ is a bilinear map from $U\times V$ to $X$.
  \item Prove that for any bilinear map $\tau\colon U\times V\to X$,
    there is a unique $A\in\Hom(U\otimes V,X)$ such that
    $\alpha=A\circ\iota$. This is the \emph{universal property} of
    the tensor product, and is summarized by the following
    commutative diagram:
    \[
    \xymatrix{ U\times V \ar[dr]_\iota\ar[rr]^\tau && X \\ &
      U\otimes V \ar@{-->}[ur]_A}
    \]
  \end{enumerate}
  
  
  
\item Use the universal property of the tensor product to prove the
  following results:
  \begin{enumerate}
  \item $U\otimes V\cong V\otimes U$ (hint: let $X=V\otimes U$);
  \item $(U\otimes V)\otimes W\cong U\otimes(V\otimes W)$;
  \item $(U\times V)\otimes W\cong (U\otimes W)\times(V\otimes W)$.
  \end{enumerate}
  
\end{enumerate}

\end{document}



