Name:
CSU ID:
Homework 9
November 6, 2015
1. The eigenvectors that we have been studying are sometimes called
right eigenvectors to distinguish them from left eigenvectors,
which are nonzero n × 1 column matrices ~x that satisfy the equation ~xT A = µ~xT for some scalar µ. What is the relationship, if any,
between the right eigenvectors and corresponding eigenvalues λ of A
and the left eigenvectors and corresponding eigenvalues µ of A? For
ease, assume A and its eigenvalues are real.
2. Solve the differential equation


4 0 1


0
~x =  2 3 2  ~x,
7 0 4


−3


~x(0) =  1 
0
3. For A below, find Ā, Re(A), Im(A), det(A), tr(A) and eigenvalues.
"
A=
−4 + i 2 + i
2
−i
#
.
4. Solve the differential equation
"
~x0 =
−1 4
−4 −2
#
"
,
~x(0) =
−2
3
#
~x
5. Consider the differential equation ~y 0 = A~y with initial condition ~y (0) =
[−3, 4, 1]T . The matrix A given below has eigenvalues λ = 2, −5, 3.
(a) For each eigenvalue, find the eigenvector by bringing λI − A to
RREF. The eigenvector should be defined with integers.
(b) Define a matrices S and D such that S −1 AS = D is a diagonal
matrix with the ordered eigenvalues given in (a).
(c) Solve the differential equation. The solution should be written as
a linear combination of eigenvectors.


4 −4 −2


47
26 
A =  −5
11 −92 −51
6. Rewrite the initial value problem y 00 + 5y 0 + 6y = 0 with initial conditions y(0) = 7, y 0 (0) = 2 in matrix/vector format.