2130_Chapter7_Notes

Check 1 deta i det a 2 ii 1 i 1 2 1i 1

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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 Theorem 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.

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