Question

Does these matrices have real or imaginary eigenvalues?

Answer 1

Can you post the matrices you are talking about?

Answer 2

OOPS. I didn't realize I posted that question. Sorry about that. As I was typing the question, I solved my own question, but I must have accidentally posted it. I'm closing the question now, but here is the matrix from the question in case you are really curious: \[ \left[\begin{matrix}a & 1 \\ 1 & a\end{matrix}\right] \], and a is a real number.

So to calculate the eigenvalues:

\[\lambda^2 -2a\lambda + (a^2 -1) = 0 \]

And, plugging that into the quadratic formula:

\[ \frac{2 \pm \sqrt{4a^2 - 4(a^2 - 1)}}{2 } \]

Which simplifies to \[a \pm1\]

So, the eigenvalues are not complex numbers...

Sorry again for the inadvertent post, but thank you for responding so that I can close the thread.

Answer 3

Oops again, the quadratic formula actually gives

\[\frac{2a\pm\sqrt{4a^2-4(a^2-1)}}{2}\]