A question about the characteristic equation

Cygni

Hi PF readers,

When trying to establish [tex]\lambda[/tex] values by solving a characteristic equation (for simplicity of 2x2 matrix) can one produce solution that contains complex roots? If yes, what does that show about the eigenvectors?

Thanks in advance!

Cygni

CFDFEAGURU

Yes, you can have complex roots. I leave the second part of your question to someone else, as I am not sure how to answer it.

Thanks
Matt

HallsofIvy

Well, it shows that the eigenvalues are complex numbers! Which means that, since the eigenvalues of a linear transformation from vector space V to itself must belong in the underlying field, if your vector space is over the real numbers that particular linear transformation does not actually have eigenvalues.

Cygni

Thanks guys, that helped.

I pretty much had an idea that if u get complex solutions in characteristic equation the eigen vectors would not exist as they'd require argand diagram in order to be displayed.

aPhilosopher

That's not quite right. If you let it operate on vectors composed of complex numbers, it would have eigenvectors and the eigenvalues of those vectors would be the roots you have found.

Cygni

Yes, but if we are dealing with cartesian coordinates, the complex roots obtained from characteristic equation would indictae that there are no possible eigenvectors in that plane. Isn't that correct?

aPhilosopher

Absolutely correct. Sorry if I missed something

HallsofIvy

It wasn't you. Cygni missed saying he was talking about vector spaces (or matrices) over the real numbers.

Cygni

Apologies for the vagueness in my question :)