complexeig.pdf - Math 54 — Complex eigenvalues and...

Math 54 — Complex eigenvalues and eigenvectors
For a real square matrix A with complex eigenvalues, the algebraic story of diagonalization parallels
the real story; but the geometry is different, in the sense that the eigenvectors have complex entries, and
cannot be represented geometrically in R
n
. What we can do instead is find a basis that reduces the matrix
to an especially nice form, representing a rotation and simultaneous scaling. Here is an example that
illustrates the result stated quickly at the end of class.
Theorem 1. Let F be a 2 × 2 matrix with non-real eigenvalues. The eigenvalues form a complex-conjugate
pair, λ
±
= p ± iq . If v
+
is an eigenvector for λ
+
, then v
-
:= ¯ v
+
is an eigenvector for λ
-
. Moreover,
Re v
+
and - Im v
+
form a basis for R
2
, and in that basis the linear transformation defined by matrix F
takes the form
M =
p - q
q p
In other words, F = RMR
- 1
if R is the matrix whose columns are Re v
+
and - Im v
-
.
Here is an example:
F =
1 . 4 0 . 5
- 0 . 4 0 . 6
; χ
F
( t ) = t
2
- 2 t + 1 . 04 , λ
±
= 1 ± 0 . 2 i, v
±
=
1
- 0 . 8 ± 0 . 4 i
.