\documentclass[12pt]{amsart}

\usepackage{241}

\title{21-241 Matrices and Linear Transformations \\
       Summer I 2012 \\
       Homework 4}

\begin{document}
  \maketitle

  \begin{defn*}
    The $n\times n$ \emph{shift} matrix is
    \[
      \begin{pmatrix}
        0 & 0 & \cdots & 0 & 0 \\
        1 & 0 & \cdots & 0 & 0 \\
        0 & 1 & \cdots & 0 & 0 \\
          & & \ddots & & \\
        0 & 0 & \cdots & 1 & 0 \\
      \end{pmatrix}
    \]
    (Often I'll call it $S$, but I won't make that a convention.)
  \end{defn*}

  \begin{defn*}
    The \emph{Hermitian} of an $m\times n$ complex matrix $A$ is the $n\times m$ matrix $A^H$ with entries $(A^H)_{ij} = \overline{A_{ji}}$.  In other words $A^H$ is the \emph{conjugate transpose} of $A$.
  \end{defn*}

  \begin{defn*}
    Suppose $m_1,\ldots,m_k$ and $n_1,\ldots,n_\ell$ are natural numbers greater than zero.  Let $m = m_1 + \cdots + m_k$ and $n = n_1 + \cdots + n_\ell$.  Let $A^{ij}$ be an $m_i$ by $n_j$ matrix, for each $i\le k$ and $j\le \ell$.  Then \emph{the block matrix with blocks $A^{ij}$} is the $m\times n$ matrix which looks like this;
    \[
      \begin{pmatrix}
        A^{11} & A^{12} & \cdots & A^{1\ell} \\
        A^{21} & A^{22} & \cdots & A^{2\ell} \\
        \vdots & \vdots &        & \vdots \\
        A^{k1} & A^{k2} & \cdots & A^{k\ell} \\
      \end{pmatrix}
    \]
  \end{defn*}

  \begin{defn*}
    Let $\SA$ be some set of $n\times n$ matrices.  (In other words, $\SA \subseteq M_n(\CC)$.)  The \emph{commutant} of $\SA$ is
    \[
      \SA' = \set{B\in M_n(\CC)}{\forall A\in \SA\; AB = BA}
    \]
  \end{defn*}

  \begin{defn*}
    An $n\times n$ matrix $U$ is \emph{upper triangular} if $U_{ij} = 0$ whenever $i > j$.  In other words, $U$ is upper triangular if it looks like this;
    \[
      \begin{pmatrix}
        U_{11} & U_{12} & U_{13} & \cdots & U_{1n} \\
          0    & U_{22} & U_{23} & \cdots & U_{2n} \\
          0    &   0    & U_{33} & \cdots & U_{3n} \\
        \vdots &        &        & \ddots & \vdots \\
          0    &   0    & \cdots &        & U_{nn} \\
      \end{pmatrix}
    \]
  \end{defn*}

  \pagebreak

  \begin{enumerate}
    \addtolength{\itemsep}{\baselineskip}
    \item  Find bases for the following subspaces.  Here $S$ denotes the $n\times n$ shift matrix.  Of course, you must show your work in each case.  [5 each]
    \begin{enumerate}
      \item  The span of
      \[
        \left\{
          \begin{pmatrix} 1 \\ 2 \\ 0 \\ 1 \\ \end{pmatrix},
          \begin{pmatrix} 2 \\ 2 \\ 2 \\ 2 \\ \end{pmatrix},
          \begin{pmatrix} 8 \\ 5 \\ -1 \\ -2 \\ \end{pmatrix},
          \begin{pmatrix} 4 \\ -1 \\ -3 \\ -6 \\ \end{pmatrix}
        \right\}
      \]
      \item\label{1.2}  $\col(A)$, where
      \[
        A = \begin{pmatrix}
           1 &     i & -i & 0 \\
          -i &     2 &  1 & 2 - i \\
           i &     1 &  3 & -1 \\
           0 & 2 + i & -1 & 4 \\
        \end{pmatrix}
      \]
      \item  $\nul(S^\top S)$.  (You should work out what $S^\top S$ is.)
      \item  $\nul(SS^\top)$.  (You should work out what $S S^\top$ is.)
    \end{enumerate}
    \item  Given each set $X$ and vector $v$ below, (i) find a subset of $X$ which is a basis for $\lspan(X)$, (ii) decide whether $v$ is in $\lspan(X)$, and (iii) find a linear combination of your basis vectors which produces $v$, if it is.  [10 each]
      \[
        \begin{matrix}
          \mathrm{(a)} &
          X = \left\{\begin{pmatrix} 1 \\ 2 \\ 3 \\ \end{pmatrix},
          \begin{pmatrix} 2 \\ 3 \\ 1 \\ \end{pmatrix},
          \begin{pmatrix} 3 \\ 2 \\ 1 \\ \end{pmatrix},
          \begin{pmatrix} 3 \\ 1 \\ 2 \\ \end{pmatrix}\right\} &
          v = \begin{pmatrix} 1 \\ 3 \\ 2 \\ \end{pmatrix} \\
          \\
          \mathrm{(b)} &
          X = \left\{\begin{pmatrix} 0 \\ 1 \\ 1 \\ 2 \\ \end{pmatrix},
          \begin{pmatrix} 1 \\ -1 \\ 1 \\ 0 \\ \end{pmatrix},
          \begin{pmatrix} 4 \\ 0 \\ 5 \\ 0 \\ \end{pmatrix}\right\} & 
          v = \begin{pmatrix} 2 \\ 2 \\ 2 \\ 2 \\ \end{pmatrix} \\
        \end{matrix}
      \]  
    \item  Find the determinant of each matrix below.  You may use your work from other problems if it's relevant.  [5 each]
    \[
      \begin{matrix}
        (a) & 
        \begin{pmatrix}
           1 &     i & -i & 0 \\
          -i &     2 &  1 & 2 - i \\
           i &     1 &  3 & -1 \\
           0 & 2 + i & -1 & 4 \\
        \end{pmatrix} \\
        \\
        (b) & 
        \begin{pmatrix}
          2 & 3 & 3 \\
          3 & 2 & 1 \\
          1 & 1 & 2 \\
        \end{pmatrix} \\
        (c) & 
        \begin{pmatrix}
           1 &  i & -1 \\
           i & -1 & -i \\
          -1 & -i &  1 \\
        \end{pmatrix} \\
      \end{matrix}
    \]
    \item  Let $A$ be an $n\times n$ complex matrix such that $A = -A^\top$.  Prove that if $n$ is odd, then $\det(A) = 0$.  [5]
    \item  Let $M = \smat{A & B \\ C & D \\}$ and $M' = \smat{A' & B' \\ C' & D' \\}$ be block matrices, where each of $A,B,C,D$ and $A',B',C',D'$ has size $n\times n$.  Prove the following formula for the product of $M$ and $M'$.  [10]
    \[
      MM' = \begin{pmatrix}
        AA' + BC' & AB' + BD' \\
        CA' + DC' & CB' + DD' \\
      \end{pmatrix}
    \]
    \item  Let $A$ be the block matrix $\smat{I_n \\ 0_{k\times n} \\}$.  Show that [10]
    \[
      A^\top A = I_n\qquad\mbox{and}\qquad AA^\top = \begin{pmatrix} I_n & 0_{n\times k} \\ 0_{k\times n} & 0_{k\times k} \\ \end{pmatrix}
    \]

    \item  Let $n\ge 1$ be a natural number.  Find the commutant of each of the following sets of matrices.
    \begin{enumerate}
      \item  $\SA = \{S\}$, where $S$ is the $n\times n$ shift matrix.  [10]
      \item  $\SB = \set{A\in M_n(\CC)}{\mbox{$A$ is invertible}}$.  [10]
      \item  The set [10]
      \[
        \SC = \set{\begin{pmatrix} A & 0 \\ 0 & A \\ \end{pmatrix}}{A\in M_n(\CC)}
      \]
      (So $\SC \subseteq M_{2n}(\CC)$.)
    \end{enumerate}
    \item  Let $A$ be an $m\times m$ matrix and $B$ an $n\times n$ matrix.  Prove that  [15]
    \[
      \det\begin{pmatrix} A & 0 \\ 0 & B \\ \end{pmatrix}
        = \det(A)\det(B)
    \]
    \item  Let $U$ be an $n\times n$ upper triangular matrix.  Prove that $\det(U) = U_{11}\cdots U_{nn}$.  [15]
  \end{enumerate}

\end{document}
