# Systems of First Order Linear Equations Part 2

• 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 ∈ R^{n} and b ∈ R^{n}

Proof. This follows from direct computation. Since for any two complex

scalars we know we have

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**

Prev | Next |