Solving homogenous system with complex eigenvaluesdxdt=2x+8ydydt=−x−2yWhen I solve the determinant of the matrix, I get λ=±2i. Then , I plug it in the matrix, and get the for the first eigenvalue, λ=2i:0=(2−2i)x+8y0=−x−(2+2i)yThis givesx=−82−2iyx=−(2+2i)y

Key to "Solving homogenous system with complex eigenvaluesdxdt=2x+8ydydt=−x−2yWhen I solve..."

Kiara Haas

2022-04-03

Solving homogenous system with complex eigenvalues

\frac{d x}{d t} = 2 x + 8 y

\frac{d y}{d t} = - x - 2 y

When I solve the determinant of the matrix, I get

λ = \pm 2 i

. Then , I plug it in the matrix, and get the for the first eigenvalue,

λ = 2 i

0 = (2 - 2 i) x + 8 y

0 = - x - (2 + 2 i) y

This gives

x = \frac{- 8}{2 - 2 i} y

x = - (2 + 2 i) y

Ireland Vaughan

Beginner2022-04-04Added 14 answers

Note that

\frac{- 8}{2 - 2 i} = \frac{- 8 (2 + 2 i)}{(2 - 2 i) (2 + 2 i)} = \frac{- 8 (2 + 2 i)}{8} = - (2 + 2 i)

,
so the two equations are saying the same thing (as they must, if the eigenvalues are correctly computed). So you can simply take, for example,

y = - 1 and x = 2 + 2 i

to get an eigenvector.

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