Math 3330
Dr. Laurel
Spring 2026
Week 13
7.5 Complex Eigenvalues
We will now talk a little about complex eigenvalues. First we will briefly review complex
numbers. Complex numbers came about as a means to solve equations of the form x2 +1 = 0,
which does not have any real-valued solutions.
Definition 1. — The complex number i is defined as i =
numbers is denoted by C = {a + ib : a, b ∈ R}.
√
−1. The set of all complex
Definition 2. — Given a complex number z = a + ib ∈ C, its complex conjugate is the
complex number z¯ = a − ib.
overline
z
Definition 3. — The modulus of a complex number z = a + ib ∈ C is the real number
√
√
if
|z| = z z¯ = a2 + b2 . Note that |zw| = |z||w| for all z, w ∈ C.
distance
away
from
origin
mathbb
right
frac
hat
left
0
R
2
b
a
czi
Definition 4. — The argument of a complex number z = a + ib ∈ C is theargiz
polar angle of
z when plotted in the complex plane, denoted by arg(z) = θ. The argument is not unique,
but it typically taken to be an angle in [0, 2π] or [−π, π].
If z = a + ib ∈ C, then
arg(z) =

tan−1 (b/a)
if z is in QI or QIV,
π + tan−1 (b/a)
if z is in QII or QIII.
Definition 5. — If z ∈ C and arg(z) = θ, then z = |z|(cos θ + i sin θ), and is called the
polar form of z.
Definition 6. — Euler’s formula is the famous identity below:1
eiθ = cos θ + i sin θ
for all θ ∈ R.
This means the the polar form of z ∈ C can also be written as
z = |z|eiθ
1
where θ = arg(z).
Euler’s formula can be proved using Maclaurin series.
Dr. Laurel
1
7.5 Notes
ath 333
frac
1i
Funfact
7.5
Example 1 Let z = 3 + 4i. Find z̄, |z|, and arg(z). Write z in polar form along with z 2
and 1/z. Write z 2 and 1/z in the form a + ib for a, b → R.
rangle
right
sqrt
frac
arg
left
tan
4
3
5
2
z1
overline
4
3
zi
rangle
theta
right
frac
left
tan
4
3
5
ez1i
right
frac
left
4
3
1
right
frac
left
tan
3
5
2
e
cz1i
an
rangle
right
cdot
left
9
6
4
3
2
y1i
4
2
7i
overline
right
cdot
frac
left
4
3
5
2
7
yz1i
BSBBU1
Example 2 We saw in Chapter that C is isomorphic to R2 . ultiplication by a complex
number is a linear transformation thanks to the distributive property. Find the matrix of the
linear transformation T : C → C given by T (z) = iz for all z → C. Describe multiplication
by i geometrically.
I
are
the standardbasis vectorsfor C
mathbb
right
cdot
left
0
T
1i
right
cdot
left
0
T
1i
left
0
1
matrix
begin
right
end
end
left
0
1
matrix
begin
right
matrix
begin
right
end
left
A
0
1
rotation by 90
counterclockwise
ath 333
7.5
Example 3 Let T : C → C be given by T (z) = (3 + 4i)z for all z ∈ C. Find the matrix of
T and describe multiplication by 3 + 4i geometrically.
right
cdot
left
4
3
T
1i
right
left
4
3
Ti
matrix
begin
right
end
left
4
3
matrix
begin
right
end
left
4
3
nicefrac
matrix
begin
right
sqrt
end
left
A
4
3
5
2
1
scaling
rotation scaling
matrix
rotation matrix by
by
right
left
3
5
yi
In fact, multiplication by a + ib always scales z ∈ C by |a + bi| =
arg(a + ib). This observation leads to De oivre s formula.
√
a2 + b2 and rotates z by
heorem 0.1 De oivre s Formula . For any n ∈ N,
(cos θ + i sin θ)n = cos(nθ) + i sin(nθ)
rangle
theta
right
left
n
ei
3
theta
right
arg
left
4
3i
tanright
frac
left
4
3
1
theta
left
n
cos not t ismright
ath 333
7.5
ometimes a characteristic polynomial fA (λ) does not have all real eros. In this case,
this leads to complex eigenvalues for a matrix, and complex eigenvectors. Despite this, the
process for diagonali ing such a matrix is the same as before.
!

3 →4
Example 3 Diagonali e the rotation-scaling matrix A =
over C.
4 3
lambda
rangle
matrix
begin
right
sqrt
frac
end
det
pm
left
9
6
0
4
3
5
2
x
1
rangle
matrix
begin
right
end
left
ker
4
3
E
yi
intra
matrix
begin
right
end
left
ker
0
4
2
x1i
sqrt
frac
pm
6
4
8
3
2i
matrix
begin
right
end
frac
ker
left
E
0
2
a
p
3
n
e
1
4
xi
s
y
k
matrix
begin
right
frac
end
left
ker
0
2
x1i
R
2
1i
matrix
begin
right
end
left
D
0
4
3
i
matrix
begin
vdots
right
end
left
S
p
n
as1i
matrix
begin
right
end
left
2
7
x1i
rangle
matrix
begin
vdots
right
cdot
end
frac
left
S
2i
1
ath 333
7.5
True False: It is possible for a 2 → 2 matrix to have one real eigenvalue and one complex
eigenvalue.
nswer:
see fact
Fact: Complex eigenvalues always come in pairs. pecifically, if λ = a + ib is a complex
¯ = a − ib is also an eigenvalue of A.
eigenvalue of A ∈ Rn×n , then λ
belowreal
If A ∈ R2 has two complex eigenvalues, then it s impossible to diagonali e A over R.
owever,
there is a way to keep things real not by diagonali
!
ing A, but by writing A = SRS where
a −b
R is a rotation-scaling matrix of the form
for a, b ∈ R.
b a
heorem 0.2.
an
n
A ∈ R2×2 a
or o A a o a
o
n a
a + ib,
n

$
A = SRS
−1
,
r S= w
" "v
!

complex
λ = a± ib
b $= 0
−1
"z = "v +iw
"
partof!z
overrightharpoon

a −b
an R =
.
b a
3 −5
partof
. Find a rotation-scaling matrix R and an invertible matrix
1 −1
−1
S such that A = SRS .
Example 4 Let A =
lambda
rangle
matrix
begin
right
sqrt
frac
end
det
pm
left
4
3
5
2
1
lambda
0
2
lambda
matrix
begin
right
frac
end
pm
left
ker
6
9
3
5
2
b
E
a1i
1i
Only need an eigenvector for lambda
right
left
X
0
2
1i
matrix
begin
right
end
left
X
2
x1i
rangle
matrix
begin
right
sqrt
end
left
ker
0
R
5
2
1i
overrightharpoon
matrix
begin
right
frac
end
left
w
0
2
1i
matrix
begin
right
end
left
0
R
S
5
2
b
a1
u
overrightharpoon
5
ath 333
7.5

2 →1 0

Example 5 Find all the eigenvalues real and complex for A = 1
3
lambda
matrix
begin
right
end
det
left
0
3
2
1
lambda
right
sqrt
frac
pm
left
0i
6
1
5
4
2
ome or for
.5 13-17,
-
2
1


0 .
1