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Unformatted text preview: ries are real numbers) and λ = a + bi is a complex eigenvalue with eigenvector v,
then v is also complex.
Furthermore: Av = λv =⇒ (Av)∗ = (λv)∗
=⇒ A∗ v∗ = λ∗ v∗ =⇒ Av∗ = λ∗ v∗
=⇒ λ∗ is also an eigenvalue for A with eigenvector v∗ .
Theorem Complex eigenvalues (and their respective eigenvectors) occur in conjugate pairs:
if λ is an eigenvalue for A with eigenvector v,
then λ∗ is also an eigenvalue for A with eigenvector v∗ .
2 / 10 How to Solve
x = veλt v = 0 is a solution of x′ = Ax
⇐⇒ λ is an eigenvalue for A with eigenvector v.
What if λ is complex?
Then veλt and v∗ eλ t are both complex solutions.
Label z = veλt = x(1) + ix(2) where x(1) , x(2) are real vectors, then
v∗ eλ t = veλt
∗ ∗ = x(1) + ix(2) =⇒ z = x(1) + ix(2) and ∗ = x(1) − ix(2) = z ∗ z ∗ = x(1) − ix(2) are solutions.
3 / 10 2 How to Solve (slide 2)
z = x(1) + ix(2) and z ∗ = x(1) − ix(2) are complex solutions.
Use z and z ∗ to...
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This note was uploaded on 03/19/2014 for the course APMA 2130 taught by Professor Bernardfulgham during the Fall '09 term at UVA.
- Fall '09