\begin{problem}{MATH 294}{FALL 1985}{FINAL}{3}{}
\prob{Find an angle $\theta$, expressed as a function of a,b, and
c so that the matrix product \[ \left( \begin{array}{cc} \cos
{\theta} & sin {\theta} \\ -\sin{\theta} & \cos{\theta}
\end{array} \right)
 \left( \begin{array}{cc}a&b\\b&c
\end{array} \right)\left( \begin{array}{cc} \cos
{\theta} & -sin {\theta} \\ \sin{\theta} & \cos{\theta}
\end{array} \right) \] is a diagonal matrix.  In particular, what is $\theta$ if $a=c$, and what is the resulting digital matrix? (Hint: $\cos{^2 \theta} - \sin{^2 \theta} = \cos{2 \theta}$; $\sin{\theta} \cos{\theta} = \frac{1}{2} \sin{2 \theta}$)}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1985}{FINAL}{5}{}
\prob{Find all of the eigenvalues of the matrix \[ \left(
\begin{array}{cccc} 0&1&1&2\\-1&0&2&3\\-1&-2&0&4\\-2&-3&-4&0 \end{array} \right)
\] Show why your answers are correct.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1985}{FINAL}{9}{}
\prob{In general, the eigenvalues of $A$ are ($A$ is a real $2
\times 2$ matrix)} \probpart{Always real.} \probpart{Always
imaginary.} \probpart{Complex conjugates.} \probpart{Either
purely real or purely imaginary.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1985}{FINAL}{10}{}
\prob{If $A$ has purely real eigenvalues, then (A is a real $2
\times 2$ matrix)} \probpart{The eigenvalues must be distinct.}
\probpart{The eigenvalues must be repeated.} \probpart{The
eigenvalues may be distinct or repeated.} \probpart{The
eigenvalues must both be zero.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1985}{FINAL}{11}{}
\prob{If $A$ has purely imaginary eigenvalues, then (A is a real
$2 \times 2$ matrix)} \probpart{The eigenvalues must have the
same magnitude but opposite sign.} \probpart{The eigenvalues must
be repeated.} \probpart{The eigenvalues may or may not be
repeated.} \probpart{The eigenvalues must both be zero.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1986}{FINAL}{3}{}
\probb{Find all eigenvalues of the matrix \[ \left[ \begin{array}{cccc}
   0 &  1 &  2 & 3 \\
  -1 &  0 &  1 & 2 \\
  -2 & -1 &  0 & 1 \\
  -3 & -2 & -1 & 0
\end{array} \right]. \]}
\probpart{For any square matrix $A$, show that if $\det{(A)}\neq 0
$, then zero cannot be an eigenvalue of $A$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1983}{FINAL}{10}{}
\prob{$A$ is the matrix given below, $\vec{v}$ is an eigenvector
of $A$. Find any eigenvalue of $A$.} \probnn{$A=\left[ \begin{array}{cccc}
  3 & 0 & 4 & 2 \\
  8 & 5 & 1 & 3 \\
  4 & 0 & 9 & 8 \\
  2 & 0 & 1 & 6
\end{array} \right]$ with $\vec{v}=$[an eigenvector of $A$]$= \left[ \begin{array}{c}
  0 \\
  2 \\
  0 \\
  0
\end{array} \right]$. }
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1984}{FINAL}{5}{}
\prob{Let $\lambda_{1}$ and $\lambda_{2}$ be distinct eigenvalues
of a matrix $A$ and let $x_1$ and $x_2$ be the associated
eigenvectors. Show that $x_1$ and $x_2$ are linearly independent.
}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1984}{FINAL}{2}{}
\prob{Find the eigenvalues and eigenvectors of the matrix \[
A=\left[  \begin{array}{ccc}
  1 & 1 & 0 \\
  1 & 1 & 1 \\
  0 & 1 & 1
\end{array}\right]
\].}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1984}{FINAL}{5}{}
\prob{Does the matrix with the zero row vector deleted have
$\lambda=0$ as an eigenvalue?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1986}{FINAL}{4}{}
\probb{Find an orthogonal matrix $R$ such that $R^T A R$ is
diagonal,where \[ A=\left[ \begin{array}{ccc}
  2 & 0 & 3 \\
  0 & 4 & 0 \\
  3 & 0 & 2
\end{array}\right]\].}
\probpart{Write the matrix $D = R^T A R$.} \probpart{What are the
eigenvectors and associated eigenvalues of $A$. }
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1987}{PRELIM 2}{4}{}
\prob{Problems (a) and (b) below concern the matrix $A$:\[ \left[ \begin{array}{ccc}
  1 &  0 &  1 \\
  0 & -1 &  4 \\
  0 &  2 & -8
\end{array}
\right]\]}\probpart{One of the eigenvalues of $A$ is 1, what are
the other(s)?}\probpart{Find an eigenvector of $A$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1987}{PRELIM 3}{4}{}
\prob{Find \textit{one} eigenvalue of the matrix $A$ below. Three
eigenvectors of the matrix are given.\[A=\left[ \begin{array}{ccccc}
   2 & 1 &  0 & -1 & 1 \\
   1 & 5 &  1 &  3 & 1 \\
   0 & 1 &  2 & -1 & 1 \\
  -1 & 3 & -1 &  5 & -1 \\
   1 & 1 &  1 & -1 & 1
\end{array}\right]\]}\probnn{The following three vectors are eigenvectors of $A$:
\[
\vec{v}_1= \left[ \begin{array}{c}
  1 \\
  1 \\
  1 \\
  -1 \\
  1
\end{array}\right],
\vec{v}_2= \left[ \begin{array}{c}
  0 \\
  2 \\
  0 \\
  2 \\
  0
\end{array}\right],
\vec{v}_3=\left[ \begin{array}{c}
  1 \\
  0 \\
  -1 \\
  0\\
  0
\end{array}\right].
\]}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{SPRING 1987}{FINAL}{10}{}
\prob{Given $A= \left[\begin{array}{cc}
  1 & 1 \\
  1 & 1
\end{array}\right]$ find $R$ so that $R A R^{-1} =\left[\begin{array}{cc}
  0 & 0 \\
  0 & 2
\end{array}\right]$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1987}{PRELIM 2}{1}{}
\prob{Find the eigenvalue and eigenvectors of the matrix $\left[ \begin{array}{cc}
  1 & 4 \\
  2 & 3
\end{array}
\right]$ }
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{FALL 1987}{PRELIM 2}{2}{}
\prob{Find the eigenvalues, eigenvectors and/or generalized
eigenvectors of the matrix $\left[ \begin{array}{cc}
  0 & 3 \\
  0 & 0
\end{array}\right]$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{89}}{\pr{2}}{1}{}
\prob{Consider the matrix \[ A = \sqbrc{\begin{array}{ccc}
  0 & 1 & 2 \\
  3 & 2 & 1 \\
  1 & 1 & 1
\end{array}} \]} \probpart{Show that $\lambda = 0$ is an eigenvalue of
$A$.}\probpart{Find a corresponding eigenvector.}
\probpart{Determine whether the system of equations \[ A \vec{x}
= \sqbrc{\begin{array}{c}
  3 \\
  3 \\
  3
\end{array}}
\] has a solution or not.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{89}}{\pr{3}}{1}{}
\prob{Consider the matrix \[ A = \sqbrc{\begin{array}{ccc}
  0 & 1 & 2 \\
  0 & 0 & 0 \\
  0 & 0 & 1
\end{array}} \]} \probpart{Show that $\lambda = 0$ is a double eigenvalue, and that $\lambda = 1$ is a simple
eigenvalue.} \probpartnn{For b) and c) below, you may use the
result of a) even if you did not show it.]} \probpart{Find all
linearly independent eigenvectors corresponding to the eigenvalues
$\lambda = 0$ and $\lambda = 1$ respectively.} \probpart{Find two
linearly independent generalized eigenvectors corresponding to
the double eigenvalue $\lambda = 0.$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{90}}{\pr{3}}{1}{}
\probb{Find the eigenvalues, eigenvectors and dimension of the
subspace of eigenvectors corresponding to each eigenvalue of \[
\sqbrc{\begin{array}{ccc}
  1  & 0  & 0 \\
  -3 & 1  & 0 \\
  4  & -7 & 1
\end{array}}
\]} \probpart{For what value of $c$ (if any) is $\lambda = 2$ an eigenvalue of \[ \sqbrc{\begin{array}{ccc}
  1  & -1 & -1 \\
  1  &  c & 1 \\
  -1 & -1 & 1
\end{array}}? \] In that case find a basis for the subspace of eigenvectors corresponding to $\lambda = 2.$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{90}}{\pr{3}}{2}{}
\probb{There is a $2 \times 2$ matrix $R$ such that $R^t A R$ is
a diagonal matrix, where $A = \sqbrc{\begin{array}{cc}
  2 & -1 \\
  -1 & 2
\end{array}}$. Find $R^t A R.$ (Hint: You needn't find $R$; there are two correct
answers)}\probpart{Describe the conic $v^t A v = 1$ for $v$ in
$V_2$ and $A = \sqbrc{\begin{array}{cc}
  4 & -1 \\
  -1 & -2
\end{array}}$. Explain why your answer is correct.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{91}}{FINAL}{3}{}
\prob{Diagonalize the one of the following matrices which can be
diagonalized: \[ A = \brc{\begin{array}{ccc}
  2 & 0 & 0 \\
  1 & 2 & 0 \\
  0 & 0 & 1
\end{array}}, B = \brc{\begin{array}{cccc}
  3 & 0 & 0 & 0 \\
  0 & 4 & 0 & 0 \\
  0 & 1 & 4 & 0 \\
  0 & 0 & 0 & 4
\end{array}}, C = \brc{\begin{array}{ccc}
  6 & -10 & 6 \\
  2 & -3 & 3 \\
  0 & 0 & 2
\end{array}}. \]}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{91}}{\pr{3}}{5}{}
\prob{Find the eigenvalues and eigenvectors of the matrix $A$
where \[ A = \brc{\begin{array}{ccc}
  3 & 4 & 2 \\
  -2 & -2 & -1 \\
  0 & -1 & 0
\end{array}}\] Hint: $\lambda = 1$ is one eigenvalue of $A$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{91}}{\pr{3}}{6}{}
\prob{An $n \times n$ matrix always has $n$ eigenvalues (some
possibly complex), but these are not always distinct. (T/F)}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{91}}{FINAL}{4}{}
\prob{Find the eigenvalues and eigenvectors of the matrix \[ A =
\brc{\begin{array}{cc}
  0 & i \\
  i & 0
\end{array}}.
\]}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{92}}{\pr{3}}{3}{}
\prob{Find the eigenvalues and three linearly independent
eigenvectors for the matrix \[ A = \sqbrc{\begin{array}{ccc}
  3 & 1 & -1 \\
  0 & 3 & 0 \\
  0 & 1 & 2
\end{array}} \]}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{92}}{FINAL}{3}{}
\prob{Consider the eigenvalue problem: Find all real numbers
$\lambda$ (eigenvalues) such that the differential equation
$-\ndpd{w}{x} = \lambda w, 0 < x < L$ with the boundary
conditions $\stpd{w}{x} (0) = \stpd{w}{x} (L) = 0$ has nontrivial
solutions (eigenfunctions). Given that there are no eigenvalues
$\lambda < 0$, find all possible eigenvalues $\lambda \geq 0$ and
corresponding eigenfunctions. You must derive your result.  No
credit will be given for simply writing down the answer. }
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{92}}{FINAL}{3}{}
\prob{A vector space $V$ has two bases \[ B_1 : \{ e^t, e^{2 t},
e^{3 t} \} \mbox{and} B_2 : {e^t + e^{2 t}, e^{3 t}, e^{2 t}} \]
A linear operator $T: V \to V$ is $T = \stpd{}{t}$}
\probpart{Find the matrix $T_{B_1}$ which represents $T$ in the
basis $B_1$.} \probpart{For the vectors $v = e^{2 t}, w =
\stpd{v}{t},$ find $\beta_1(v)$ and $\beta_1(w)$ which represent
these vectors in the basis $B_1.$} \probpart{Noting that $T(v) =
2 v, $ i.e. $v$ is an eigenvector of $T$ with eigenvalue equal to
2, interpret the equation \[ \beta_1(w) = T_{B_1} \beta_1 (v) \]
as an eigenvalue-eigenvector equation for $T_{B_1}$. What are the
eigenvalue and eigenvector in this equation? } \probpart{Now
consider the basis $B_2$. Find the matrices $(B_2 : B_1)$ and
$(B_1 : B_2)$.} \probpart{Find $\beta_2{v}, T_{B_2}$ and
$\beta_2(w).$} \probpart{Is the equation $\beta_2(w) = T_{B_2}
\beta_2(v)$ also an eigenvalue-eigenvector equation? If so, what
are the eigenvalue and eigenvector in this case?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{92}}{FINAL}{4}{}
\probb{Find the eigenvalues and eigenvectors of the matrix \[ B =
\sqbrc{\begin{array}{cc}
  1 & 1 \\
  -1 & 1
\end{array}}.
\]} \probpart{Let $A = \sqbrc{\begin{array}{cc}
  4 & 2 \\
  -1 & 1
\end{array}}$. Find a nonsingular matrix $C$ such that $C^{-1} A C = D$ where $D$ is a diagonal matrix.  Find $C^{-1}$ and $D$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{92}}{FINAL}{6}{}
\prob{Find the eigenvalues and eigenfunctions (nontrivial
solutions) of the two-point boundary-value problem \[ y^{\prime
\prime} + \lambda y = 0, 0 < x < 1, \mbox{(assume} \lambda \geq 0)
\] \[ y^{\prime}(0) = y(1) = 0. \]}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{93}}{FINAL}{4}{}
\prob{It is known that a $3 \times 3$ matrix $A$ has: 1) A
twice-repeated eigenvalue $\lambda_1 = 1$ with corresponding
eigenvectors $v_1 = \brc{\begin{array}{c}
  1 \\
  1 \\
  0
\end{array}}$
and $v_2 = \brc{\begin{array}{c}
  1 \\
  0 \\
  1
\end{array}}$, and 2) another eigenvalue $\lambda_2 = 0$ with corresponding eigenvectors $v_3 = \brc{\begin{array}{c}
  0 \\
  1 \\
  1
\end{array}}$} \probpart{Find a basis for the null space $Nul(B)$ of
$B$, where $B$ is the matrix $B=(A-I_3$, and $I_3$ is the $3
\times 3$ identity matrix.}\probpart{Find a basis for the null
space $Nul(A)$ of $A$.} \probpart{For part a) above, $Null(B)$ is
a plane in 3-dimensional space. The equation of this plane can be
written in the form $A x + B y + C z = 0.$ Find $A, B, $ and
$C.$} \probpart{For part $b$ above, is $Null(A)$ a line, a plane,
or something else? Please explain your answer carefully. }
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{94}}{FINAL}{5}{}
\prob{Let $A = \sqbrc{\begin{array}{ccc}
  -3 & 0 & -4 \\
  0 & 5 & 0 \\
  -4 & 0 & 3
\end{array}}$.} \probpart{Find the eigenvalues of $A$.}
\probpart{Find a basis for the eigenspace associated with each
eigenvalue. The eigenspace corresponding to an eigenvalue is the
set of all eigenvectors associated with the eigenvalue, plus the
zero vector.} \probpart{Find an orthogonal matrix $P$ and a
diagonal matrix $D$ so that $P^{-1} A P = D$. What is $D$?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{94}}{FINAL}{7}{}
\prob{If an $n \times n \; A$ has $n$ distinct eigenvalues, then}
\probpart{$\det(A)$ not zero,} \probpart{$\det(A)$ is zero,}
\probpart{$A$ is similar to $I_n$,} \probpart{$A$ has $n$
linearly independent eigenvectors,} \probpart{$A = A^T$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{94}}{FINAL}{10}{}
\prob{Which of the following is an eigenvector of
$\sqbrc{\begin{array}{cc}
  1 & 1 \\
  0 & 0
\end{array}}$?} \probnn{a. $\vttwo{2}{-1}$, b. $\vttwo{0}{1}$, c. $\vttwo{0}{0}$, d. $\vttwo{1}{1}$, e. $\vttwo{-1}{1}$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{95}}{FINAL}{6}{}
\prob{Let \[ A = \mttwo{0}{1}{-1}{0}. \]} \probpart{Find all
eigenvalues of $A$, and for each an eigenvector.} \probpart{Find
a matrix $P$ such that $D = P^{-1} A P$ is diagonal.}
\probpart{Find $D$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{95}}{FINAL}{7}{}
\probb{Find all eigenvalues of the matrix \[
\mtthree{-2}{1}{0}{1}{-2}{1}{0}{1}{-2}.
\]} \probpart{For each eigenvalue find a corresponding
eigenvector.}\probpart{Are the eigenvectors orthogonal?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{95}}{\pr{3}}{5}{}
\probb{One eigenvalue of $A = \brc{\begin{array}{cc}
  3 & 1 \\
  5 & 7
\end{array}}$ is 2. Find a corresponding eigenvector.} \probpart{Find the characteristic polynomial $\det(A-\lambda I)$ if $A = \brc{\begin{array}{cccc}
  6 & 0 & 0 & 0 \\
  0 & 3 & 1 & 0 \\
  0 & 5 & 7 & 0 \\
  0 & 0 & 0 & \pi
\end{array}}$. Also find all the eigenvalues of $A$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{95}}{FINAL}{6}{}
\prob{For $A = \mtthree{0}{-2}{1}{-2}{0}{-1}{1}{-1}{1}$}
\probpart{Show that the characteristic polynomial of $A$ is \[
-\lambda (\lambda^2 - \lambda - 6)\]} \probpart{Find all
eigenvalues of $A$ of 3 linearly independent eigenvectors.}
\probpart{Check your solution of part (b).}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\fa{95}}{FINAL}{8}{}
\prob{For $A = \sqbrc{\artwo{0}{2}{2}{3}}$} \probpart{Find the
eigenvalues and 2 linearly independent eigenvectors. Show that
these 2 eigenvectors are orthogonal.} \probpart{Find a matrix $P$
and a diagonal matrix $D$ so that $P^{-1} A P = D$} \probpart{If
your matrix $P$ in part (b) is not orthogonal, how can it be
modified to make it orthogonal? (so that $P^{-1} A P = D$ still
holds).}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{96}}{\pr{3}}{5}{}
\prob{Let \[ B = \brc{\arthree{2}{0}{0}{0}{1}{4}{0}{4}{1}}. \]}
\probpart{Find the characteristic polynomial of $B$.}
\probpart{Find the eigenvalues of $B$. Hint: one eigenvalue is
2.} \probpart{Find eigenvectors corresponding to the eigenvalues
other than 2.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{96}}{\pr{3}}{6}{}
\prob{Let $C$ be a 2-by-2 matrix. Suppose \[ \vec{v} =
\vttwo{1}{1} \mbox{ and } \vec{w} = \vttwo{1}{-1}
\] are eigenvectors for $C$ with eigenvalues 1 and 0, respectively. Let \[ \vec{x} = \vttwo{6}{12}. \] Find $C^{100} \vec{x}.$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{96}}{FINAL}{14}{}
\prob{Let $A = \brc{\artwo{a}{b}{-b}{a}}, a, b \in \Re.$ A
complex eigenvector of $A$ is:} \probnn{a) $\vttwo{-2}{i}$ b)
$\vttwo{2}{-i}$ c) $\vttwo{1}{i}$ d) $\vttwo{-i}{2}$ e) none of
the above}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{}{}{5}{}
\prob{}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{96}}{FINAL}{15}{}
\prob{Let \[ A = \brc{\arthree{3}{1}{0}{1}{3}{0}{0}{0}{2}}.\]
Then} \probpart{$A$ has three linearly independent eigenvectors
with eigenvalue 2.} \probpart{The eigenspace corresponding to the
eigenvalue 2 has a basis consisting of exactly one eigenvector.}
\probpart{The eigenspace corresponding to the eigenvalue 2 has
dimension 2.} \probpart{$A$ is not diagonalizable.}
\probpart{None of the above.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{96}}{FINAL}{38}{}
\prob{The only matrix with 1 as an eigenvalue is the identity
matrix. (T/F)}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{96}}{FINAL}{39}{}
\prob{If $A$ is an $n \times n$ matrix for which $A = P D
P^{-1}$, $D$ diagonal, then $A$ cannot have $n$ linearly
independent eigenvectors. (T/F)}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{96}}{FINAL}{40}{}
\prob{If $x$ is an eigenvector of a matrix $A$ corresponding to
the eigenvalue $\lambda$, then $A^3 \vec{x} = \lambda^3 \vec{x}.$
(T/F) }
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{97}}{\pr{2}}{2}{}
\prob{Let $A = \sqbrc{\arthree{9}{0}{0}{1}{0}{-2}{1}{2}{0}},
\vec{b} = \vtthree{1}{1}{0},$ and $\vec{x} =
\vtthree{x_1}{x_2}{x_3}$} \probpart{Find the characteristic
polynomial $\det(A-\lambda I)$ of $A$, and find all the
eigenvalues. (\emph{hint}: $\lambda-9$ is one factor of the
polynomial.} \probpart{Find an eigenvector for each eigenvalue.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{97}}{FINAL}{5}{}
\prob{Let \[ A = \sqbrc{\arthree{1}{0}{0}{0}{1}{-1}{0}{-1}{1}}
\]} \probpart{Find the characteristic polynomial of $A$. Verify that the eigenvalues of $A$ are: 0, 1,
2}\probpart{For each eigenvalue, find a basis for the
corresponding eigenspace.} \probpart{Find a diagonal matrix $D$
and an invertible matrix $P$ such that $A = P D P^{-1}$}
\probpart{Show  that the columns of $P$ form an orthogonal basis
for $\Re^3$.} \probpart{Find $A^{10} \vec{x}$ where $\vec{x} =
\vtthree{1}{1}{1}$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{97}}{\pr{2}}{4}{}
\prob{The following information is known about a $3 \times 3$
matrix $A$.  (Here $e_1, e_2, e_3$ is the standard basis for
$\Re^3$).} \probparts{$A e_1 = \vtthree{1}{1}{2},$}
\probparts{$e_1 + e_2$ is an eigenvector of $A$ with
corresponding eigenvalue 1.} \probparts{$(e_2 + e_3)$ is an
eigenvector of $A$ with corresponding eigenvalue 2.} \probnn{Find
the matrix $A$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{97}}{\pr{3}}{3}{}
\prob{Given an $n \times n$ matrix $A$ with $n$ linearly
independent eigenvectors, it is possible to find a square root of
$A$ (that is, an $n \times n$ matrix $\sqrt{A}$) with
$\brc{\sqrt{A}^2 = A})$ by using the following method:}
\probnn{(1) Find $D = P^{-1} A P,$ where $D$ is a diagonal matrix
(and $P$ is some suitable matrix).} \probnn{(2) Find $\sqrt{D}$ by
taking the square roots of the entries on the diagonal. (This
might involve complex numbers).} \probnn{(3) The square root of
$A$ is then $\sqrt{A} = P \sqrt{D} P^{-1}.$} \probnn{Use this
method to find $sqrt{A}$ for the matrix \[ A =
\brc{\artwo{0}{1}{1}{0}}.
\]}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{97}}{FINAL}{3}{}
\prob{Let  \[ A = \brc{\begin{array}{cccc}
  0  & -1 & 0 & 0 \\
  -1 & 0  & 0 & 0 \\
  0  & 0  & 0 & 1 \\
  0  & 0  & 1 & 0
\end{array}} \] Find the eigenvalues and eigenvectors of $A$. Is $A$ diagonalizable?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{97}}{\pr{3}}{4}{}
\probb{Let \[ A =
\brc{\artwo{\cos\theta}{\sin\theta}{\sin\theta}{\cos\theta}}.
\] For which values of $\theta$ is this matrix diagonalizable?}
\probpart{Let $A$ be $2 \times 2$ matrix with characteristic
polynomial $(\lambda-1)^2$. Suppose that $A$ is diagonalizable.
Find a matrix $A$ with these properties. Now, find \emph{all}
possible matrices $A$ with these properties. Justify your answer!}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{97}}{\pr{3}}{5}{}
\prob{Let \[ A = \sqbrc{\arthree{1}{0}{0}{0}{1}{-1}{0}{-1}{1}}
\]} \probpart{Find the characteristic polynomial of $A$. Verify that the eigenvalues of $A$ are: 0, 1,
2}\probpart{For each eigenvalue, find a basis for the
corresponding eigenspace.} \probpart{Find a diagonal matrix $D$
and an invertible matrix $P$ such that $A = P D P^{-1}$}
\probpart{Show  that the columns of $P$ form an orthogonal basis
for $\Re^3$.} \probpart{Find $A^{10} \vec{x}$ where $\vec{x} =
\vtthree{1}{1}{1}$}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{98}}{\pr{1}}{2}{}
\prob{Find the values of $\lambda$ (eigenvalues) for which the
problem below has a non-trivial solution. Also determine the
corresponding non-trivial solutions (eigenfunctions.) \[
y^{\prime \prime} + \lambda y = 0 \mbox{ for } 0 < x < 1
\] \[ y(0) = 0, y^{\prime}(1) = 0. \] (Hint: $\lambda$ must be positive for non-trivial solutions to exists. You may assume this.)}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{98}}{\pr{2}}{5}{}
\prob{Consider the matrix $A$: \[ A =
\sqbrc{\arthree{2}{1}{1}{0}{1}{0}{-1}{-1}{2}}
\] Find \emph{all} the eigenvalues of $A$ and find a corresponding eigenvector for \emph{each} eigenvalue. (Hint: 1 is an eigenvalue.)}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{98}}{\pr{3}}{4}{}
\prob{Let $A = \sqbrc{\artwo{2}{1}{1}{2}}$} \probpart{Find the
eigenvalues and eigenvectors of $A$.} \probpart{Diagonalize $A$.
That is, give $P$ and $D$ such $A = P D P^{-1}$.} \probpart{Let
$\vec{e}_1 = \vttwo{1}{0}$ and $\vec{e}_2 = \vttwo{0}{1}$ be the
standard basis vectors of $\Re^2$. Map $\vec{e}_1 \to P\vec{e}_1$
and $\vec{e}_2 \to P\vec{e}_2$ and sketch $P \vec{e}_1$ and $P
\vec{e}_2$.} \probpart{Give a geometric interpretation of
$\vec{x} \to P \vec{x}$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{98}}{\pr{3}}{5}{}
\prob{True or false? Justify each answer.} \probpart{In general,
if a finite set $S$ of nonzero vectors spans a vector space $V$,
then some subset of $S$ is a basis of $V$.} \probpart{A linearly
independent set in a subspace $H$ is a basis for $H$.}
\probpart{An $n \times n$ matrix $A$ is diagonalizable if and only
if $A$ has $n$ eigenvalues, counting multiplicities.}
\probpart{If an $n \times n$ matrix $A$ is diagonalizable, it is
invertible.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{98}}{\pr{3}}{3}{}
\probb{} \probpart{$\vec{v} = \vtfour{1}{2}{3}{-1}$ is an
eigenvector of $A = \sqbrc{\begin{array}{cccc}
  10 & -4 & 6  & 5  \\
  -4 &  8 & 4  & -6  \\
  6  &  4 & 10 & -1 \\
  5  & -6 & -1 & 5
\end{array}}$. Find an eigenvalue of $A$.} \probpart{Consider the $20 \times 20$ matrix that is all zeros but for the main diagonal. The main diagonal has the numbers 1 to 20 in order. Precisely describe as many eigenvalues and eigenvectors of this matrix as you
can.}\probpart{If possible, diagonalize the matrix $A =
\sqbrc{\artwo{-2}{6}{6}{7}}$. Explicitly evaluate any relevant
matrices (if any inverses are needed the can be left in the form
$[\;]^{-1}$).}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{98}}{FINAL}{6}{}
\prob{Let $A = \sqbrc{\artwo{1}{1}{1}{1}}$.} \probpart{Find
orthonormal eigenvectors $\{\vec{v}_1, \vec{v}_2\}$ of $A$.
[Hint: do not go on to parts d-e below until you have double
checked that you have found two orthogonal unit vectors that are
eigenvectors of $A$.]} \probpart{Use the eigenvectors above to
diagonalize $A$.} \probpart{Make a clear sketch that shows the
standard basis vectors $\{ \vec{e}_1, \vec{e}_2 \}$ of $\Re^2$
and the eigenvectors $\{ \vec{v}_1, \vec{v}_2 \}$ of $A$.}
\probpart{Give a geometric interpretation of the change of
coordinates matrix, $P$, that maps coordinates of a vector with
respect to the eigen basis to coordinates with respect to the
standard basis.} \probpart{Let $\vec{b} = \vttwo{3}{5}.$ Using
orthogonal projection express $\vec{b}$ in terms of $\{
\vec{v}_1, \vec{v}_2 \}$ the eigenvectors of $A$.}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{93}}{FINAL}{6}{}
\probb{Write a matrix $A = S^{-1}{A}{S}$ such that $A$ is
diagonal, if \[ A =
\brc{\arthree{6}{-10}{6}{2}{-3}{3}{0}{0}{2}}\]} \probpart{What is
matrix $S$?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{93}}{FINAL}{6}{}
\probb{Write a matrix $\Lambda = S^{-1} A {S}$ such that
$\Lambda$ is diagonal, if \[ A =
\brc{\arthree{6}{-10}{6}{2}{-3}{3}{0}{0}{2}}\]} \probpart{What is
the matrix $S$? }
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{\spr{98}}{\pr{2}}{5}{}
\prob{Consider the matrix $A$: \[ A =
\sqbrc{\arthree{2}{1}{1}{0}{1}{0}{-1}{-1}{2}}\] Find all the
eigenvalues of $A$ and find a corresponding eigenvector for each
eigenvalue. (Hint: 1 is an eigenvalue.)}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{99}}{\pr{2}}{2b}{}
\prob{The matrix $A = \sqbrc{\begin{array}{cccc}
  1 & 3 & 4 & 2.718 \\
  3 & 5 & \pi & 8 \\
  \sqrt{5} & 3 & 4 & 1 \\
  3 & 4 & 6 & 7
\end{array}}$ has 4 distinct eigenvalues (no eigenvalue is equal to any of the others. The eigenvectors of $A$ make up the 4 columns of a matrix
$P$. Does the equation $P \vec{x} =
\vtfour{2.718}{\pi}{\sqrt{5}}{4}$ have} \probpart{no solution
(why?), or} \probpart{exactly one solution (why?), or}
\probpart{exactly two solutions (why?), or}\probpart{an infinite
number of solutions (why?), or}\probpart{it depends on
information not given (what information?, how would that
information tell you the answer?)?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{99}}{\pr{2}}{2c}{}
\prob{The matrix $A = \sqbrc{\begin{array}{cccc}
  66 & -52 & 8 & -4 \\
  -52 & 83 & -26 & -24 \\
  8 & -26 & 54 & -52 \\
  -4 & -24 & -52 & 22
\end{array}}$ has four eigenvalues $\lambda_1 = -30, \lambda_2 = 30, \lambda_3 = 90, \lambda_4 = 135.$ Some four vectors $\vec{v}_i$ satisfy $A \vec{v}_i = \lambda_i \vec{v}_i$ (for i = 1,2,3,4). This is all you are told about the vectors $\vec{v}_i$. The vectors $\vec{v}_i$ make up, in the order given, the columns of a matrix $P$. If this is sufficient to answer each of the questions below, then answer the questions, if not explain why you need more information. No credit for unjustified correct answers. [Hint: massive amounts of arithmetic are not needed for any of the three
parts].}\probpart{What is the element in the third row and second
column of $P^T P$?} \probpart{What is the element in the third
row and third column of $P^{-1} A P$?} \probpart{What is the
element in the second row and second column of $P^T P$?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\spr{99}}{\pr{2}}{3}{}
\prob{A couch potato spends \emph{all} of his/her time either
smoking a cigarette ('C') \emph{or} eating a bag of fries ('F')
\emph{or} watching a TV show ('T'). Since there is no smoking
inside and no food allowed in the living room, he/she only does
one thing at a time. \begin{itemize} \item After a cigarette
there is a 50\% chance he/she will light up again, but right
after smoking he/she never eats, (If you think this is ambiguous
please reread the initial statement,) \item After eating a bag of
fries he/she has a 50\% chance of going out for a smoke, a 25\%
chance of eating another bag of fries, and a 25\% chance of
turning the TV on. \item After watching TV show he/she only wants
to eat.
\end{itemize} On average he/she watches 300 TV shows a month. \textbf{On average, how many cigarettes does he/she smoke in a month?}}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{UNKNOWN}{FINAL}{8}{}
\probb{Let $A$ be a nonsingular $n \times n$ matrix, $X, B n
\times n$ matrices. Solve the equation $\sqbrc{A \times A^T}^T -
B^T $ for $X$ and show that $X$ is symmetric.} \probpart{Let
$\vec{u} = \sqbrc{u_1, \ldots, u_n}^T$ and $C = I - \vu \vu^T.$
Express the entries $c_{i j}$ of the matrix $C$ in terms of the
$u_i$. Show that $C$ has zero as an eigenvalue provided that
$||\vu|| =1.$ Determine the corresponding \underline{unit}
eigenvector. (Hint: Do not attempt to evaluate the characteristic
polynomial of $C$. Use instead the definition of eigenvalue and
eigenvector.) }
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SPRING ?}{FINAL}{4}{}
\probb{Determine the real numbers a,b,c,d,e,f, given that
$\brc{\arthree{1}{1}{1}{a}{b}{c}{d}{e}{f}}$ has eigenvectors
$\vtthree{1}{1}{1}, \; \vtthree{1}{0}{-1}, \;
\vtthree{1}{-1}{0}$} \probpart{What are the corresponding
eigenvalues?}
\end{problem}
%----------------------------------
\begin{problem}{MATH 294}{\fa{87}}{\pr{3}}{1 PRELIM}{}
\probb{Find the general solution of the system $\vec{x}^\prime =
A \vec{x} $ if \[ A = \brc{\begin{array}{cccc}
  2 & 1 & 0 & 0 \\
  0 & 2 & 1 & 0 \\
  0 & 0 & 2 & 1 \\
  0 & 0 & 0 & 1
\end{array}} \and \vec{x} = \vtfour{x_1}{x_2}{x_3}{x_4}\]}
\probpart{How many independent eigenvectors can we find for the
matrix \[ \brc{\begin{array}{cccc}
  2 & 1 & 0 & 0 \\
  0 & 2 & 1 & 0 \\
  0 & 0 & 2 & 1 \\
  0 & 0 & 0 & 1
\end{array}}? \]}
\end{problem}
%----------------------------------
\begin{problem}{MATH 293}{SUMMER 1992}{FINAL}{2}{}
\prob{Given \[A = \frac{1}{2}
\brc{\arthree{3}{0}{1}{1}{4}{-1}{1}{0}{3}}
\]} \probpart{Find all the eigenvalues of $A$.} \probpart{Find all linearly independent eigenvectors of
$A$.}\probpart{Can $A$ be diagonalized by a change of basis? If
so, let $D = (B:S)^{-1} A(B:S)$ where $D$ is diagonal.}
\probnn{Find $(B:S) \and D.$}
\end{problem}
