12.4: Review Problems
( \newcommand{\kernel}{\mathrm{null}\,}\)
1. Try to find more solutions to the vibrating string problem ∂2y/∂t2=∂2y/∂x2 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=(2102). Find all eigenvalues of M. Does M have two linearly independent} eigenvectors? Is there a basis in which the matrix of M is diagonal? i.e., can M be diagonalized?)
3. Consider L:ℜ2→ℜ2 with L(xy)=(xcosθ+ysinθ−xsinθ+ycosθ).
a) Write the matrix of L in the basis (10),(01).
b) When θ≠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)=λv.
e) Are there complex eigenvalues for L, assuming that i=√−1 exists?
4. Let L be the linear transformation L:ℜ3→ℜ3 given by
L(xyz)=(x+yx+zy+z).
Let ei be the vector with a one in the ith position and zeros in all other positions.
a) Find Lei for each i.
b) Given a matrix M=(m11m12m13m21m22m23m31m32m33), what can you say about Mei for each i?
c) Find a 3×3 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 λ. Show that v is also an eigenvector for A2 and what is its eigenvalue? How about for An where n∈N? Suppose that A is invertible. Show that v is also an eigenvector for A−1.
6. A projection is a linear operator P such that P2=P. Let v be an eigenvector with eigenvalue λ for a projection P, what are all possible values of λ? 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 any field.
7. Explain why the characteristic polynomial of an n×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 general explanation.
8. Compute the characteristic polynomial PM(λ) of the matrix M=(abcd). Now, since we can evaluate polynomials on square matrices, we can plug M into its characteristic polynomial and find the matrixPM(M). What do you find from this computation?
Does something similar hold for 3×3 matrices? (Try assuming that the matrix of $M$ is diagonal to answer this.)
9. Discrete dynamical system. Let M be the matrix given by M=(3223).
Given any vector v(0)=(x(0)y(0)), we can create an infinite sequence of vectors v(1),v(2),v(3), and so on using the rule:
v(t+1)=Mv(t) 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)=⋯ (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),… 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)