hw18
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moienMath 21b: Linear Algebra Spring 2016
Homework 18: Discrete Dynamical systems
This homework is due on Wednesday, March 23, respectively on Thursday, March 24, 2016.
10 12
1 Show that has the eigenvalues 20 and 4 Find the corre-
5 14
sponding eigenvectors.
6 8
2 a) The matrix A = is a reflection dilation. Use geometric
8 −6
insight to find the eigenvalues
and
eigenvectors of A.
3 0 −4
b) The matrix B = 0 2 0 is a rotation dilation on the xz-
4 0 3
plane and a dilation in the y-axes. Use this to find an eigenvector
and eigenvalue.
3 A Lilac bush has n(t) new branches and o(t) old branches at the
beginning of each year t. During the year, each old branch will
grow two new branches and every new branch will become
a old
n(t + 1) n(t)
branch. a) Find the matrix A such that = A .
o(t + 1) o(t)
1 2
b) Verify that v1 = and v2 = are eigenvectors. Find
1 −1
the eigenvalues.
n(0)
c) Find closed formulas for n(t), o(t) if the initial condition =
c(0)
c1v1 + c2v2 are given. If you prefer to work with an example, take
4
the initial condition .
1
2 1 0
4 a) Find the characteristic polynomial of A = 1 2 1 and de-
0 1 2
termine its roots.
b) What are the eigenvalues and eigenvectors of the projection
P (x, y, z) = (x, y, 0) from space to the xy-plane?
5 a) Find the characteristic polynomial of 5 × 5 matrix for which
the diagonal entries are Akk = k and Akl = 0 with k 6= l.
b) Why does every 11 × 11 matrix have a real eigenvalue.
c) Find a 6 × 6 matrix for which there is no real eigenvalue.
Eigenvalues
A nonzero vector v is an eigenvector of A, if Av = λv for some real number λ called
eigenvalue. A basis B consisting of eigenvectors of A is called an eigenbasis. Eigenvalues
λj and vectors vj help to solve discrete dynamical systems x → Ax, where we want
to find closed formulas for the trajectories At x: write an initial vector x as a sum of
eigenvectors x = c1 v1 + ... + cn vn , then get At x = c1 λt1 v1 + ... + cn λtn vn . One can find
eigenvalues as roots of the characteristic polynomial fA (λ) = det(A − λIn ). It is a
polynomial of degree n of the form
fA (λ) = (−λ)n + tr(A)(−λ)n−1 + .... + det(A) .
The
algebraic multiplicity
of an eigenvalue is the multiplicity of the root. The matrix
1 1 1
A= 1 1 1
for example has the characteristic polynomial
1 1 1
1−λ 1 1
fA (λ) = det(
1 1−λ 1 )
1 1 1−λ
which is −λ3 +3λ2 = λ2 (3−λ) showing that λ = 0 is an eigenvalue of algebraic multiplicity
2 and λ = 3 is an eigenvalue of multiplicity 1.