Math 308
Worksheet 5: Sections 7.2 - 7.3
18 & 20 July 2012
Solving a system of two equations in two unknowns
Key Idea. Start with a system of two equations in two unknowns
(1a)
(1b)
ax + by = 0
cx + dy = 0 .
Notice that (x, y) = (0, 0) is always a solution of (1). We write
0
∈ SOL(1) .
0
The point (b, −a) is a solution of the first equation (1a). It will also be a solution
of the second equation (1b) if
(2)
0 = cb − da .
This tells us: the second equation (1b) is redundant if (2) holds.
Remark. A second useful way to think about (2) is to rewrite it as −a/b = −c/d.
This is precisely the statement that the two lines (1a) and (1b) have the same slope.
Since they both have y–intersept 0, (2) is telling us: they are the same line!
Case 1: 0 6= ad − bc. Then (x, y) = (0, 0) is the only solution to (1). We write
0
SOL(1) =
.
0
Case 2: 0 = ad − bc. Then every point of the form (x, y) = (tb, −ta) = t(b, −a) is
a solution of (1). We write
b
t∈R .
SOL(1) = t
−a Key Idea. The system (1) can be expressed as a single matrix equation:
a b
x
ax + by
0
dfn
(3)
=
=
.
c d
y
cx + dy
0
Write
A =
a b
c d
.
Then
0
det(A) 6= 0 =⇒ SOL(3) =
,
0
b
det(A) = 0 =⇒ SOL(3) = t
−a
t∈R .
Eigenvalue and Eigenspace Summary
Definition. A vector V =
v1
v2
is an eigenvector of the matrix A =
a b
c d
, with
eigenvalue λ if AV = λV .
Remark. If V is an eigenvector with eigenvalue λ, then so is tV , for any number t ∈ R
you choose. The eigenspace E(λ) is the set of all the eigenvectors of V . (In most
cases, E(λ) is the line {tV | t ∈ R} spanned by V .)
a−λ
b
To find the eigenvalues λ of A. Solve the equation 0 = det
.
c
d−λ
To find the eigenspaceE(λ).
an eigenvalue
Select
λ, that
you found
above.
To
0
a−λ
b
v1
v1
find the eigenspace, solve
=
for V =
.
0
c
d−λ
v2
v2
Remark. Be aware that eigenvectors are not unique. When you solve for V , you will
get a one-paramater space of solutions. This reflects the fact: if V is an eigenvector,
then so is tV for any number t ∈ R you choose.
Exercise 1. Compute the eigenvalues and eigenspaces for each of the matrices below.
5 4
1 0
−2 3
6 −5
2 3
−2 3
4 5
9 −5
2 2
4 −4
9 −3
−1 4
7 −3
1 1
3
1
.
3 1
−4 5
−4 −1
Exercise 2. Use the information
5
10
A
=
1
2
and A
1
−2
−3
6
−14
2
=
to determine the eigenvalues and eigenspaces of A.
Exercise 3. Use the information
1
5
A
=
3
15
and A
7
−1
to determine the eigenvalues and eigenspaces of A.
Complex Eigenvalues
=
When the eigenvalues λ = α ± iβ of the matrix A are complex (β 6= 0), the
eigenvectors will also be complex. If we write an eigenvector V of one eigenvalue
α + iβ as
r1 + is1
r1
s1
V =
=
+ i
= R + iS ,
r2 + is2
r2
s2
then R − iS is an eigenvector for the second eigenvalue α − iβ. Moreover, the eigenequation AV = λV tells us that
AR = αR − βS
Exercise 4. Use the information
3
6
6
A
=
−
2
4
3
and AS = βR + αS .
and A
2
1
=
9
6
+
4
2
.
to determine the eigenvalues and eigenspaces of A.
Exercise 5. Use the information
5
15
2
A
=
−
1
3
4
and A
1
2
=
10
2
+
3
6
.
to determine the eigenvalues and eigenspaces of A.
Generalized Eigenvectors
When the matrix A has only one eigenvalue λ, then we may be in one of two
situations:
λ 0
. Then every vector is an eigenCase 1. The first possibility is that A =
0 λ v1
vector; that is, the eigenspace is E(λ) =
v1 , v2 ∈ R all vectors.
v2
λ 0
Case 2. In this case, A 6=
, and the eigenspace E(λ) is a single eigen-line.
0 λ
In this case, we look for generalized eigenvectors.
w1
Definition. If λ is an eigenvalue of A with eigenvector V , then W =
is a
w2
generalized eigenvector for the pair λ, A if
AW = V + λW .
To find generalized eigenvectors W for λ, V . Solve
w1
for W =
.
w2
v1
v2
=
a−λ
b
c
d−λ
w1
w2
Remark. Be aware that (like eigenvectors) generalized eigenvectors are not unique.
When you solve for W you will get a one-paramater space of solutions. This reflects
the fact: if W is a generalized eigenvector, then so is W + tV for any number t ∈ R
you choose.
Exercise 6. The third row of matrices in Exercise 1 consists of matrices with a single
eigenvalue: compute generalized eigenvectors each matrix.
Exercise 7. Use the information
3
6
A
=
2
4
2
1
and A
=
7
4
.
to determine the eigenvalues, eigenspaces and generalized eigenvectors of A.
Exercise 8. Use the information
5
15
=
A
1
3
1
2
and A
8
7
=
.
to determine the eigenvalues, eigenspaces and generalized eigenvectors of A.
Exercise 9. Use the information
2
1
A
=
4
2
−2
0
and A
0
2
=
.
to determine the eigenvalues, eigenspaces and generalized eigenvectors of A.
Exercise 10. Use the information
1
3
A
=
12
−24
and A
1
2
=
−3
−6
.
to determine the eigenvalues, eigenspaces and generalized eigenvectors of A.
Exercise 11. Use the information
7
0
A
=
3
0
and A
4
0
=
7
3
.
to determine the eigenvalues, eigenspaces and generalized eigenvectors of A.
Homework. As part of your preparation for Quiz 3 (Wednesday, July 25) and the
Final Exam (Tuesday, August 07), I recommend the following:
(A) Section 7.2: Problems 22, 23, 25.
(B) Section 7.3: Problems 16-18.
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Colleen Robles
robles@math.tamu.edu