[Math] Complex eigenvalues of a matrix in conjugate pairs (or not)

Question

I have learnt that in a matrix, if there are complex eigenvalues, they should come as conjugate pairs. Also, I know that, in a diagonal matrix, eigenvalues are the diagonal elements.

So how about the following matrix?

$$\begin{pmatrix}
i & 0\\
0& 2
\end{pmatrix}$$

Shouldn't the eigenvalues be $i$ and $2$, where it doesn't have a conjugate pair?!

I appreciate your help to clarify my mistake.

Answer 1

Complex eigenvalues of matrices with real entries come as conjugate pairs.

This is not necessarily the case for matrices with complex entries.