\begin{center}
MA464, Matrix Theory \\
Dr. E. A. Coutsias \\
Final Examination:Part I\\[.2in]
\end{center}
Fall 1990 \\[.2in]
{\em Instructions:} This part of the exam is due in class
on Friday, Dec. 7. You can use the textbook, your notes and 
your wits.  You should not discuss the problems with anyone 
else, in or outside the class. 
\begin{enumerate}

        \item 
Suppose $x\in {\cal{R}}^n$ with $x_k \neq 0$. If
\[ \alpha_i = x_i/x_k \qquad\qquad\qquad i=k+1,\ldots ,n   \]
and
\[ M_k = I -\alpha e_k^T, \qquad\qquad \alpha^T=(0,\ldots,0,\alpha_{k+1},\ldots,\alpha_n) \]
(with $e_k$ the k-th column of the identity matrix $I$),
then
\[ M_k \left[ \begin{array}{l}
               x_1\\ \vdots \\ x_k \\ x_{k+1} \\ \vdots \\ x_n
        \end{array} \right]=
       \left[ \begin{array}{l}
               x_1\\ \vdots \\ x_k \\ 0 \\ \vdots \\ 0
        \end{array} \right]. \]
The matrix $M_k$ is said to be a Gauss transformation. Show that\\
\begin{enumerate}
\item
  If $M_k = I - \alpha e_k^T$ is a Gauss transformation, then $M_k^{-1} = I+\alpha e_k^T$.
\item
  Suppose $\alpha^{(k)} = (0,\ldots,0,\alpha_{k+1,k},\ldots,\alpha_{nk})^T\in{\cal{R}}^n$
for $k=1,\ldots,n-1$. Show that if $M_k = I - \alpha^{(k)}e_k^T$ and 
$L = (M_{n-1}\cdots M_1)^{-1}$, then $l_{ij} = \alpha_{ij}$ for all $i > j$.
\item
  Determine a $3$--by--$3$ Gauss transformation $M$ such that
\[ M\left[ \begin{array}{r} 2 \\ 3 \\ 4 \end{array} \right] =
    \left[ \begin{array}{r} 2 \\ 7 \\ 8 \end{array} \right] . \]
\end{enumerate}
%\newpage
        \item
\begin{enumerate}
        \item
 Show that if the $n\times n$ matrix $A$ is symmetric positive definite, 
then there exists a lower triangular $n\times n$ matrix $L$  with positive
diagonal entries such that  $ A = L L^T$ (Cholesky factorization).
        \item 
Find the Cholesky factorization for the matrix
\[ A = \left[ \begin{array}{rr}
                   4 & 12 \\
                  12 & 45    \end{array} \right] .\]
        \item
Employ the Cholesky factorization to 
find a least squares solution to the inconsistent system
\[  \left[ \begin{array}{rr}
           1 & 2 \\ 3 & 4 \\ 5 & 6 \end{array} \right] x =
    \left[ \begin{array}{rr}
           1 \\ 1 \\ 1 \end{array} \right]  . \]
\end{enumerate}
%\newpage
       \item
\begin{enumerate}
       \item
Suppose that the $n\times n$ matrix $J$ is a single Jordan canonical
block, with diagonal entries equal to $\lambda$. Show that the solution 
to the system of differential equations
\[ {\bf y}^{\prime} = J {\bf y} \ \ \ , \ \ \  {\bf y}(0) = {\bf y}_0 \]
can be written in the form
\[ {\bf y}(t) = K {\bf y}_0 \]
where
\[ K = e^{\lambda t} \left[
     \begin{array}{rrrrr}
  1 & t & \frac{1}{2}t^2 & \cdots & \frac{t^{n-1}}{(n-1)!} \\
  0 & 1 & t              & \cdots & \frac{t^{n-2}}{(n-2)!} \\
    &   &                & \cdots &                        \\
  0 & 0 & 0              & \cdots &  1                     \end{array} \right] . \]
       \item
Show that the result in (i) can be extended to general Jordan form matrices.

       \item
Use the result in (i) to solve the system
\[ \frac{d{\bf z}}{dt} = \left[ \begin{array}{rrr}
      5 & 4 & 3 \\ -1 & 0 & -3 \\ 1 & -2 & 1 \end{array} \right] {\bf z} .\]
\end{enumerate}
%  \newpage
       \item
A Givens rotation is an orthogonal matrix of the form
\[ J_{ik}(\theta) =
\left[ \begin{array}{rrrrr}
                     1     & \vdots &       &\vdots &        \\
                    \cdots &    c   & \cdots &   s   & \cdots \\
                           & \vdots &       &\vdots &        \\
                    \cdots &   -s   & \cdots &   c   & \cdots \\
                           & \vdots &       &\vdots &    1   
\end{array}\right]. \]
\begin{enumerate}
        \item 
Show that one can write $c$ and $s$ as $c=\cos \theta$ and $s = \sin \theta$
for some angle $\theta$.
        \item 
What is the geometric effect of a Givens rotation 
$J_{ik}(\theta)$ applied to ${\cal{R}}^n$?
        \item
Show that one can choose a Givens rotation to map the vector
$x=(1,2,3,4)^T$ into a vector $y$ of the form $y=(--,--,--,0)^T$.
        \item
Show how one can choose a Givens rotation R to accomplish the
transformation
\[ R^T\left[ \begin{array}{rrr}
             4 & 4 & 2 \\
             4 & 4 & 1 \\
             2 & 1 & 8  \end{array} \right]R =
      \left[ \begin{array}{rrr}
                 4     & 2\sqrt{5} & 0   \\
             2\sqrt{5} &    5.6    & 2.2 \\
                 0     &    2.2    & 6.4  \end{array} \right] .\]
\end{enumerate}
\end{enumerate}
