MAT 315: TEST 3.
Problem 1. Note that in this problem the matrix is not diagonalizable and has
only two different eigenvalues. It is a multipart problem in which we consider the
following 3 × 3-matrix

2

A= 0
−4
1
2
−12

0

0 .
3
Part A. Find the two eignvalues λ1 and λ2 and fill out the following table
Eigenvalues
λ1 =
λ2 =
AlgebraicM ultiplicity
alg.mult(λ1 ) =
alg.mult(λ2 ) =
Eigenvectors
vλ1 =
vλ2 =
GeometricM ultiplicity
geo.mult(λ1 )
geo.mult(λ2 )
Part B. Use the table above to determine the Jordan normal form J of the
matrix A.
Part C. One of the eigenvalues above has a different algebraic multiplicity then
geometric multiplicity. This means we need to find a generalized eigenvector q. To
this aim, use the matrix equation AQ = QJ to find this vector q.
Part D. Calculate eJt and uses this to calculate eAt = QeJt Q−1 by using your
answer in Part C.
Part E. (Extra credit) Consider the non-homogouns system of differential equad
tions dt
v = Av + b(t) where b(t) = (1, t, 0). Calculate the particular soloution
R
At
vp = e u(t) by first computing u(t) = e−tA b(t)dt + C where C = (c1 , c2 , c3 ).
Part F. (Extra credit) Use the initial condition v(0) = (1, 2, −1)T to solve the
initial value problem in Part E and write down the resulting particular solution vp .
Problem 2. Find the Jordan"normal decomposition
A = QJQ−1 to calculate
#
√
13 −19
.
A of the following matrix A =
9 −48
1
2
MAT 315: TEST 3.
Problem 3. Consider the following matrix


2 1 0


A = 0 1 5 .
0 −2 3
Part A. Calculate the two Jordan blocks J1 and J2 of A. One of them is a
complex Jordan block, say J2
"
cos(θ)
Part B. Write J2 in polar coordinate form J2 = r2 ·
− sin(θ)
sin(θ)
cos(θ)
#
Part C. Find the eigenvectors of A and put them in a transition matrix Q with
real number entries.
Part D. How does A = QJQ−1 act on the vector space R3 .
Problem 4 Briefly explain why one of the following facts are true.
A) Every symmetric m atrix is diagonalizable.
B) Every linear system of homogenous linear ordinary differential equations has
a unique solution.
C) The determinant of a matrix is the product of all of its eigenvalues counting
multipilicity.
D) Every n × n matrix has at least one eigenvector.
2
E) The standard nomral curve f (x) = √12π e−x is an eigenfunction of the Fourier
R∞
transform F(f (x))(s) = −∞ f (x)e−isx dx.
Problem 5
Part A. Calculate N 0 , N 1 , N 2 , N 3 of the

0 1

N = 0 0
0 0
following matrix

0

1
0
Part B. Calculate the matrix polynomial p(t) = I + N t + N 2 t2 /2 + N 3 t3 /6
Part C. Calculate (2I + N )0 , (2I + N )1 , (2I + N )2 /2, (2I + N )3 /6
Part D. Approximate eJt where J = 2I + N by calculating
by using Part C.
P3
k=0 (2I
+ N )k /k!
Part E. Verify directly the claim in Part D by calculating eJt by using the
formula for exponentials of matrices.