Show two matrices are conjugate in $GL_2(\mathbb{C})$ Let $GL_2(\mathbb{C})$ be the group of invertible two-by-two matrices over the complex numbers, with matrix multiplication the group operation.
Question: Show that $\begin{bmatrix}0 & 1 \\ -1 & 0\end{bmatrix}$ is conjugate to $\begin{bmatrix}i & 0 \\ 0 & -i\end{bmatrix}$, 
ie. there exists $A \in GL_2(\mathbb{C})$ such that $A \begin{bmatrix}0 & 1 \\ -1 & 0\end{bmatrix} A^{-1} = \begin{bmatrix}i & 0 \\ 0 & -i\end{bmatrix}$.
This might be silly, but I can't seem to figure out how to cause the 'change in orientation' when I am conjugating. I have tried conjugating with $\begin{bmatrix}0 & 1 \\ i & 0\end{bmatrix}$ and $\begin{bmatrix}1 & 0 \\ 0 & i\end{bmatrix}$, and neither of these work. 
Ideally, I'd like to know not just the solution, but why the solution works!

Notice that the second given matrix is diagonal, so you can think of it as the result of diagonalizing the first one. So all you've got to do is use the standard procedure for diagonalizing the first matrix: you already know its eigenvalues (the common eigenvalues of both matrices are clearly seen in the second one), and you only need to find the corresponding eigenvectors.
