Diff EQ Repeated Complex Eigenvalues?

[th3ownly]

1. What dimensions of a matrix will give repeated complex Eigenvalues? Give an example
of one and show that it has repeated complex Eigenvalues.


2. No really equations needed?

The Attempt at a Solution

My attempt is a 2x2 which i don't think is right but here it is.

If the matrix were
we will use x as lambda

(x)' = [2+3i - λ 0 ] (x)
(y) ' = [ 0 2+3i-λ] (y)
This would yield the eigenvalues 2+3i, 2+3i


I just don't think my solution is right

 

[Dick]

If you mean the matrix [[2+3i,0],[0,2+3i]] has a double complex eigenvalue of 2+3i, I don't see what could be wrong with that.

 

[Feldoh]

Why do you think it doesn't? In order for a non-trivial solution det(A-λI) = 0 right?

Do it out -- does that equation ever equal zero?

However I think there is a better way to go about doing this you've simply listed a property of a 2x2 matrix where the eigenvalues will be the diagonal elements as long as everything else is 0.

The characteristic equation of a 2x2 matrix:

[A-λ, B]
[C, D-λ]

Is a quadratic, right?

How do we know if the roots will be imaginary from looking at the characteristic equation?

 

[th3ownly]

Why do you think it doesn't? In order for a non-trivial solution det(A-λI) = 0 right?

Do it out -- does that equation ever equal zero?

However I think there is a better way to go about doing this you've simply listed a property of a 2x2 matrix where the eigenvalues will be the diagonal elements as long as everything else is 0.

The characteristic equation of a 2x2 matrix:

[A-λ, B]
[C, D-λ]

Is a quadratic, right?

How do we know if the roots will be imaginary from looking at the characteristic equation?

necause there is no sign change?