

\documentclass[12pt]{amsart}

\usepackage{241}

\title{21-241 Matrices and Linear Transformations \\
       Summer I 2012 \\
       Homework 6}

\begin{document}

 \maketitle

  \begin{defn*}
    A square, \emph{complex} matrix $P$ is called a \emph{projection matrix} if $P^2 = P = P^H$.  (Note the difference with the earlier definition of projection.)
  \end{defn*}

  \begin{defn*}
    A matrix $A\in M_n(\CC)$ is \emph{diagonalizable} if there are some diagonal $D$ and invertible $S$ such that $A = SDS^{-1}$.  In this case we say $S$ \emph{diagonalizes} $A$.
  \end{defn*}

  \begin{defn*}
    A matrix $U\in M_n(\CC)$ such that $U^H U = I$ and $U U^H = I$ (ie, $U$ is invertible and $U^{-1} = U^H$) is called \emph{unitary}.  If $U\in M_n(\RR)$ and $U$ is unitary, we sometimes say that $U$ is \emph{orthogonal}.
  \end{defn*}

  The following theorem may be useful in the problems below.
  \begin{thm*}
    If $A$ is a Hermitian matrix, then there is some unitary $U$ such that $UAU^H$ is diagonal, and moreover the diagonal values of $UAU^H$ are exactly the eigenvalues of $A$.
  \end{thm*}

  You may also use the following characterization of the determinant (often called the Laplace expansion).
  \begin{fact*}
    Let $A$ be an $n\times n$ matrix with entries $a_{ij}$.  Then for any $i$ or $j$,
    \[
      \det(A) = \sum_{k=1}^n (-1)^{i+k} a_{ik} \det(M_{ik}) = \sum_{k=1}^n (-1)^{k+j} a_{kj} \det(M_{kj})
    \]
    where $M_{ij}$ is the $(n-1)\times (n-1)$ matrix obtained from $A$ by removing its $i$th row and $j$th column entirely.  (This is the so-called \emph{$(i,j)$-minor} of $A$.)
  \end{fact*}
  \vfill

  \pagebreak

\begin{enumerate}
\addtolength{\itemsep}{\baselineskip}

  \item  Given each matrix $A$ below, check whether $A$ is diagonalizable; if it is, diagonalize it.  Otherwise, demonstrate an eigenvalue of $A$ whose geometric multiplicity is different from its algebraic multiplicity.  [10 each]
  \[
    \begin{matrix}
      \\
      (a) & \begin{pmatrix}
        1 & 1 & 1 & 1 \\
        0 & 1 & 1 & 1 \\
        0 & 0 & 1 & 1 \\
        0 & 0 & 0 & 1 \\
      \end{pmatrix} &  &
      (b) & \begin{pmatrix}
         1 &  i & -1 & -i \\
        -i &  1 &  i & -1 \\
        -1 & -i &  1 &  i \\
         i & -1 & -i &  1 \\
      \end{pmatrix} \\
      \\
      (c) & \begin{pmatrix}
        2 & 0 & 2 & 0 \\
        0 & 2 & 0 & 2 \\
        2 & 0 & 2 & 0 \\
        0 & 2 & 0 & 2 \\
      \end{pmatrix} &  &
      (d) & \begin{pmatrix}
         1 & i & 0 \\
        -i & 1 & i \\
        0 & -i & 1 \\
      \end{pmatrix} \\
      \\
      (e) & \begin{pmatrix}
        0 & 0 & 1 \\
        1 & 0 & 0 \\
        0 & 1 & 0 \\
      \end{pmatrix} &  & 
      (f) & \begin{pmatrix}
        1 & 1 \\
        1 & 0 \\
      \end{pmatrix}
    \end{matrix}
  \]
  \item  Use your answer from 1(f) to prove that
  \[
    f_n = \frac{1}{\sqrt{5}}(\phi_+^n - \phi_-^n)
  \]
  where $f_n$ is the $n$th Fibonacci number, and
  \[
    \phi_\pm = \frac{1 \pm \sqrt{5}}{2}
  \]
  (Recall from previous homework that if $A = \smat{1 & 1 \\ 1 & 0}$ then  $A^n = \smat{f_{n+1} & f_n \\ f_n & f_{n-1}}$.)  [10]

  \item  Show that there is an $A\in M_2(\RR)$ such that $\det(A)\ge 0$, but there is no $B\in M_2(\RR)$ such that $A = B^2$.  [10]

  \item  Show that if $A\in M_n(\CC)$ is Hermitian and $\spec(A)\subseteq [0,\infty)$, then there is some $B\in M_n(\CC)$ such that $A = B^2$.  [10]
  \item  Let $P$ be a projection matrix.  Show that $\tr(P) = \rank(P)$.  [10]

  \item  We call a matrix $U\in M_n(\CC)$ \emph{distance-preserving} if $\norm{Ux - Uy} = \norm{x - y}$ for all $x,y\in\CC^n$.  Prove that the following are equivalent;  [20]
    \begin{enumerate}
      \item\label{dist}  $U$ is distance-preserving.
      \item\label{norm}  For all $x\in\CC^n$, $\norm{Ux} = \norm{x}$.
      \item\label{ip}  For all $x,y\in\CC^n$, $\brak{Ux}{Uy} = \brak{x}{y}$.
      \item\label{unit}  $U$ is unitary.
    \end{enumerate}
    (Hint: for the $\eqref{norm}\implies\eqref{ip}$ implication, consider the quantities $\norm{Ux - Uy}^2$ and $\norm{Ux - U(iy)}^2$.)

  \item  Prove that if $U$ is an orthogonal matrix then every \emph{real} eigenvalue of $U$ is either $+1$ or $-1$, and either $\det(U) = +1$ or $\det(U) = -1$.  [10]
  \item  Let $U$ be a $2\times 2$ orthogonal matrix.  Prove that $U$ is either a rotation matrix or a reflection matrix.  [10]
  \item  Let $n$ be an odd natural number.  Prove that every $A\in M_n(\RR)$ has at least one real eigenvalue.  [10]
  \item  Let $U$ be a $3\times 3$ orthogonal matrix.
    \begin{enumerate}
      \item  Prove that there is some unit vector $v\in\RR^3$ such that either $Uv = v$ or $Uv = -v$.  [10]
      \item  Prove that there are vectors $t,u\in\RR^3$ such that $\{t,u,v\}$ is an orthonormal basis of $\RR^3$.  [5]
      \item  Let $\hat{U}$ be the matrix of $T_U$ with respect to the (ordered) basis $(t,u,v)$.  Show that $\hat{U}$ has one of the following forms;  [15]
      \[
      \begin{matrix}
        \\
        \begin{pmatrix}
          \cos\theta & -\sin\theta & 0 \\
          \sin\theta &  \cos\theta & 0 \\
              0      &      0      & 1 \\
        \end{pmatrix} &  &
        \begin{pmatrix}
          \cos(2\theta) &  \sin(2\theta) & 0 \\
          \sin(2\theta) & -\cos(2\theta) & 0 \\
              0      &      0      & 1 \\
        \end{pmatrix} \\
        \\
        \begin{pmatrix}
          \cos\theta & -\sin\theta &  0 \\
          \sin\theta &  \cos\theta &  0 \\
              0      &      0      & -1 \\
        \end{pmatrix} &  &
        \begin{pmatrix}
          \cos(2\theta) &  \sin(2\theta) & 0 \\
          \sin(2\theta) & -\cos(2\theta) & 0 \\
              0      &      0      & -1 \\
        \end{pmatrix} \\
        \\
      \end{matrix}
      \]
      \item  Describe the linear transformations $T_U$ and $T_{\hat{U}}$, in geometric terms, in each of the cases above.  (You may write down a single description of what's going on, so long as it covers all the cases, and isn't ambiguous.)  [10]
    \end{enumerate}

  \item  Let $S$ be a subspace of $\CC^n$ and let $x\in\CC^n$.  Prove that the distance between $\PP_S(x)$ and $x$ is the smallest possible distance between $x$ and any vector in $S$.  Stated more formally, prove that
  \[
    \norm{x - \PP_S(x)} \le \norm{x - s}
   \]
   for all $s\in S$.  (Hint; note that $x - s = (x - \PP_S(x)) + (\PP_S(x) - s)$.)  [10]

  \item  Let $S$ be the $n\times n$ shift matrix.  Find a polynomial in $S$ and $S^\top$ which evaluates to  [10]
    \[
      \begin{pmatrix}
        0 & & & & \\
          & 1 & & & \\
          & & 2 & & \\
          & & & \ddots & \\
          & & & & n-1 \\
      \end{pmatrix}
    \]

\end{enumerate}

\end{document}
