
\documentclass[12pt]{amsart}

\usepackage{241}

\title{21-241 Matrices and Linear Transformations \\
       Summer I 2012 \\
       Homework 5}

\begin{document}
 \maketitle

  \begin{defn*}
    If $V$ is a subspace of $\CC^n$, we define the \emph{orthogonal subspace of $V$} to be
    \[
      V^\perp = \set{x\in\CC^n}{\forall v\in V\; x\perp v}
    \]
    We say that subspaces $V$ and $W$ of $\CC^n$ are \emph{orthogonal}, and write $V\perp W$, if
    \[
      \forall v\in V\;\forall w\in W\quad v\perp w
    \]
  \end{defn*}

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

   \item  Find an orthonormal basis for each of the following subspaces.  [10 each]
   \[
     \begin{matrix}
       (a) & \lspan\left\{
         \begin{pmatrix}
           1 \\ 0 \\ 2 \\ 2 \\
          \end{pmatrix},
         \begin{pmatrix}
           0 \\ 2 \\ 1 \\ 5 \\
          \end{pmatrix}
          \right\} \\
        \\
      (b) & \col\begin{pmatrix}
           0      & -2      & 1 \\
           3 - 2i &  0      & i \\
          -1      & -i      & 0 \\
           0      &  1      & 2 \\
           1 +  i &  1 -  i & 0 \\
          \end{pmatrix}


     \end{matrix}
   \]
   \item  Find the eigenvalues of the following matrices.  For each eigenvalue, find an orthonormal basis for its associated eigenspace.  (Feel free, in finding the roots of the characteristic polynomial, to use a calculator.)  
   \[
     \begin{matrix}
       (a) & & \begin{pmatrix}
            1 & 1 \\
            1 & 0 \\
          \end{pmatrix} & & [5] \\
     \end{matrix}
    \]
    \pagebreak
    \[
     \begin{matrix}
       (b) & \begin{pmatrix}
            0 &  i & -1 \\
           -i &  0 &  i \\
           -1 & -i &  0 \\
          \end{pmatrix} & [10] \\
        \\
       (c) & \begin{pmatrix}
            2 & 0 & 2 & 0 \\
            0 & 2 & 0 & 2 \\
            2 & 0 & 2 & 0 \\
            0 & 2 & 0 & 2 \\
          \end{pmatrix} & [10] \\
        \\
       (d) & \mbox{The $n\times n$ shift matrix.} & [10] \\
     \end{matrix}
  \]

  \item  Let $P$ and $Q$ be $n\times n$ projections.  Show that the following are equivalent; [20]
    \begin{enumerate}[(i)]
      \item  $PQ = 0$.
      \item  $\ran(P) \perp \ran(Q)$.
      \item  $P + Q$ is a projection.
    \end{enumerate}

  \item  Prove or disprove: $\rank(AB) = \rank(BA)$ for all square matrices $A$ and $B$ of the same size.  [10]

  \item  Prove that for all $x,y\in\CC^n$ and $\lambda\in\CC$,
    \begin{enumerate}
      \item  If $x\perp y$ then $\norm{x + y}^2 = \norm{x}^2 + \norm{y}^2$.  [5]
      \item  $\norm{\lambda x} = |\lambda| \norm{x}$.  [5]
      \item  $\norm{x + y} \le \norm{x} + \norm{y}$.  (Hint: use the Cauchy-Schwartz inequality.)  [10]
    \end{enumerate}

  \item  Let $S$ and $T$ be subspaces of $\CC^n$, and suppose $S\perp T$.  Prove that $S\cap T = \{0\}$.  [5]

  \item  Let $A$ be a matrix satisfying $A = A^H$.  Prove that $\spec(A)\subseteq\RR$.  (Hint: look at the number $\brak{Av}{v}$, where $v$ is an eigenvector of $A$.)  [10]

\end{enumerate}

\end{document}
