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6. Let
A
=
•
a
b
c
d
‚
be a real matrix with non–real eigenvalues
λ
=
a
+
ib
and
λ
=
a

ib
, with corresponding eigenvectors
X
=
U
+
iV
and
X
=
U

iV
,
where
U
and
V
are real vectors. Also let
P
be the real matrix defned by
P
= [
U

V
]. Finally let
a
+
ib
=
re
iθ
, where
r >
0 and
θ
is real.
(a) As
X
is an eigenvector corresponding to the eigenvalue
λ
, we have
AX
=
λX
and hence
A
(
U
+
iV
)
=
(
a
+
ib
)(
U
+
iV
)
AU
+
iAV
=
aU

bV
+
i
(
bU
+
aV
)
.
Equating real and imaginary parts then gives
AU
=
aU

bV
AV
=
bU
+
aV.
(b)
AP
=
A
[
U

V
] = [
AU

AV
] = [
aU

bV

bU
+
aV
] = [
U

V
]
•
a
b

b
a
‚
=
P
•
a
b

b
a
‚
.
Hence, as
P
can be shown to be non–singular,
P

1
AP
=
•
a
b

b
a
‚
.
(The ±act that
P
is non–singular is easily proved by showing the columns o±
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This note was uploaded on 12/19/2011 for the course MAS 3105 taught by Professor Dreibelbis during the Fall '10 term at UNF.
 Fall '10
 Dreibelbis

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