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Systems of First Order Linear Equations Part 2

Complex Eigenvalues

• If a real matrix A has a complex eigenvalue then
must also be an eigenvalue.

• This follows from the fact that det(A − λI) is a polynomial of
degree
n with real coefficients .

• We want real valued solutions to

x′ = Ax

• The eigenvectors of A corresponding to λ and will be complex .

Eigenvectors of Conjugate Eigenvalues

Lemma 21.1. If the matrix has conjugate eigenvalues
and with eigenvectors and
respectively then

where a ∈ Rn and b ∈ Rn

Proof. This follows from direct computation. Since for any two complex
scalars we know we have

Solutions

If can be shown that u(t) and v(t) are linearly independent solutions to
the ODE.

If the matrix A ∈ has conjugate eigenvalues and
with eigenvectors and and
distinct eigenvalues with real eigenvectors then a
general solution of x ′ = Ax is

The result is straightforward to adapt to multiple conjugate pairs and
distinct real eigenvalues .

Example

Textbook p. 401

x′ = Ax

Example

z1

Example


 

Example

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