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COMPLEX EIGENVALUES
Math 21b, O. Knill
HOMEWORK: Section 7.5: 8,12,24,32,38,60*,36*
NOTATION. Complex numbers are written as
z
=
x
+
iy
=
r
exp(
iφ
) =
r
cos(
φ
) +
ir
sin(
φ
).
The real
number
r
=

z

is called the
absolute value
of
z
,
the value
φ
is the
argument
and denoted by arg(
z
).
Complex numbers contain the
real numbers
z
=
x
+
i
0 as a subset. One writes Re(
z
) =
x
and Im(
z
) =
y
if
z
=
x
+
iy
.
ARITHMETIC. Complex numbers are added like vectors:
x
+
iy
+
u
+
iv
= (
x
+
u
) +
i
(
y
+
v
) and multiplied as
z
*
w
= (
x
+
iy
)(
u
+
iv
) =
xu

yv
+
i
(
yu

xv
). If
z
6
= 0, one can divide 1
/z
= 1
/
(
x
+
iy
) = (
x

iy
)
/
(
x
2
+
y
2
).
ABSOLUTE VALUE AND ARGUMENT. The absolute value

z

=
p
x
2
+
y
2
satisFes

zw

=

z
 
w

.
The
argument satisFes arg(
zw
) = arg(
z
) + arg(
w
).
These are direct consequences of the polar representation
z
=
r
exp(
iφ
)
, w
=
s
exp(
iψ
)
, zw
=
rs
exp(
i
(
φ
+
ψ
)).
GEOMETRIC INTERPRETATION. If
z
=
x
+
iy
is written as a vector
±
x
y
²
, then multiplication with an
other complex number
w
is a
dilationrotation
: a scaling by

w

and a rotation by arg(
w
).
THE DE MOIVRE ±ORMULA.
z
n
= exp(
inφ
) = cos(
nφ
) +
i
sin(
nφ
) = (cos(
φ
) +
i
sin(
φ
))
n
follows directly
from
z
= exp(
iφ
) but it is magic:
it leads for example to formulas like cos(3
φ
) = cos(
φ
)
3

3 cos(
φ
) sin
2
(
φ
)
which would be more di²cult to come by using geometrical or power series arguments. This formula is useful
for example in integration problems like
R
cos(
x
)
3
dx
, which can be solved by using the above deMoivre formula.
THE UNIT CIRCLE. Complex numbers of length 1 have the form
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 Spring '03
 JUDSON
 Linear Algebra, Algebra, Complex Numbers

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