
12.4: Review Problems

1.    Try to find more solutions to the vibrating string problem $$\partial^{2} y/\partial t^{2}=\partial^{2} y/\partial x^{2}$$
using the ansatz
$$y(x,t)=\sin(\omega t) f(x)\, .$$
What equation must $$f(x)$$ obey? Can you write this as an eigenvector equation? Suppose that the string has length $$L$$ and $$f(0)=f(L)=0$$.  Can you find any solutions for $$f(x)$$?

2.    Let $$M=\begin{pmatrix} 2 & 1 \\ 0 & 2 \end{pmatrix}$$.  Find all eigenvalues of $$M$$.  Does $$M$$ have two linearly independent} eigenvectors?  Is there a basis in which the matrix of $$M$$ is diagonal? $$\textit{i.e.}$$, can $$M$$ be diagonalized?)

3.    Consider $$L \colon \Re^{2}\rightarrow \Re^{2}$$ with $$L\begin{pmatrix}x\\y\end{pmatrix} =\begin{pmatrix}x\cos \theta +y\sin \theta\\ -x\sin \theta + y\cos \theta\end{pmatrix}\, .$$
a)    Write the matrix of $$L$$ in the basis $$\begin{pmatrix}1\\0\end{pmatrix}, \begin{pmatrix}0\\1\end{pmatrix}$$.

b)    When $$\theta\neq 0$$, explain how $$L$$ acts on the plane.  Draw a picture.

c)    Do you expect $$L$$ to have invariant directions?

d)    Try to find real eigenvalues for $$L$$ by solving the equation
$L(v)=\lambda v.$

e)    Are there complex eigenvalues for $$L$$, assuming that $$i=\sqrt{-1}$$ exists?

4.    Let $$L$$ be the linear transformation $$L \colon \Re^{3}\rightarrow \Re^{3}$$ given by
$$L\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix} x+y \\ x+z \\ y+z\end{pmatrix}.$$
Let $$e_{i}$$ be the vector with a one in the $$i$$th position and zeros in all other positions.

a)    Find $$Le_{i}$$ for each $$i$$.

b)    Given a matrix $$M=\begin{pmatrix} m^{1}_{1} & m^{1}_{2} & m^{1}_{3}\\ m^{2}_{1} & m^{2}_{2} & m^{2}_{3}\\ m^{3}_{1} & m^{3}_{2} & m^{3}_{3}\\ \end{pmatrix}$$, what can you say about $$Me_{i}$$ for each $$i$$?

c)    Find a $$3\times 3$$ matrix $$M$$ representing $$L$$.  Choose three nonzero vectors pointing in different directions and show that $$Mv=Lv$$ for each of your choices.

d)    Find the eigenvectors and eigenvalues of $$M.$$

5.    Let $$A$$ be a matrix with eigenvector $$v$$ with eigenvalue $$\lambda$$. Show that $$v$$ is also an eigenvector for $$A^{2}$$ and what is its eigenvalue? How about for $$A^{n}$$ where $$n \in \mathbb{N}$$? Suppose that $$A$$ is invertible. Show that $$v$$ is also an eigenvector for $$A^{-1}$$.

6.    A $$\textit{projection}$$ is a linear operator $$P$$ such that $$P^{2} = P$$. Let $$v$$ be an eigenvector with eigenvalue $$\lambda$$ for a projection $$P$$, what are all possible values of $$\lambda$$? Show that every projection $$P$$ has at least one eigenvector.

Note that every complex matrix has at least 1 eigenvector, but you need to prove the above for $$\textit{any}$$ field.

7.    Explain why the characteristic polynomial of an $$n\times n$$ matrix has degree $$n$$.  Make your explanation easy to read by starting with some simple examples, and then use properties of the determinant to give a $$\textit{general}$$ explanation.

8.    Compute the characteristic polynomial $$P_{M}(\lambda)$$ of the matrix $$M=\begin{pmatrix} a & b \\ c & d \end{pmatrix}\, .$$  Now, since we can evaluate polynomials on square matrices, we can plug $$M$$ into its characteristic polynomial and find the $$\textit{matrix} P_{M}(M)$$.  What do you find from this computation?
Does something similar hold for $$3 \times 3$$ matrices? (Try assuming that the matrix of $M$ is diagonal to answer this.)

9.    $$\textit{Discrete dynamical system.}$$ Let $$M$$ be the matrix given by $M= \begin{pmatrix} 3 & 2 \\ 2 & 3 \\ \end{pmatrix}.$
Given any vector $$v(0)= \begin{pmatrix} x(0) \\ y(0) \\ \end{pmatrix},$$ we can create an infinite sequence of vectors $$v(1), v(2), v(3),$$ and so on using the rule:
$v(t+1)=M v(t) \text{ for all natural numbers } t.$ (This is known as a {\it discrete dynamical system} whose {\it initial condition} is $$v(0).$$)

a)    Find all eigenvectors and eigenvalues of $$M.$$

b)    Find all vectors $$v(0)$$ such that $v(0)=v(1)=v(2)=v(3)=\cdots$ (Such a vector is known as a {\it fixed point} of the dynamical system.)

c)    Find all vectors $$v(0)$$ such that $$v(0), v(1), v(2), v(3), \ldots$$ all point in the same direction. (Any such vector describes an {\it invariant curve} of the dynamical system.)