12.4: Review Problems

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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.)

Contributor

David Cherney, Tom Denton, and Andrew Waldron (UC Davis)