\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}