Why is this change of basis useful? In my textbook there is a theorem which states

Let $A$ be a real $2\times 2$ matrix with complex eigenvalues $\lambda =a\pm bi$ (where $(b\ne 0)$.  If $\mathbf x$ is an eigenvector of $A$ corresponding to $\lambda=a-bi$, then the matrix $P=\begin{bmatrix} \operatorname{Re}(\mathbf x) & \operatorname{Im}(\mathbf x)\end{bmatrix}$ is inveritble and $$A=P\begin{bmatrix} a & -b \\ b & a\end{bmatrix}P^{-1}$$

My question is: why is this important?  I understand why diagonalizing a matrix is important -- it's easier to operate on diagonal matrices than arbitrary matrices.  But why is this decomposition (is this a decomposition?  I've heard the word and it sounds like what this is) useful at all?  Does it have something to do with the fact the complex numbers can be represented by $2\times 2$ matrices?

Edit: I'm also curious if this has any analogs in higher dimensions.  That is, can we find a similar decomposition for $3\times 3$ (or $4\times 4$ or $5\times 5$) matrices?
Thanks.

You can write $$C=\alpha\left[\begin{array}{cc}\cos\theta&-\sin\theta\\ \sin\theta &\cos\theta\end{array}\right]$$
$C^n$ involves $n\theta$ so it is easy to calculate, and hence powers of $A$ are easy to calculate.
On the other hand, I don't know how useful it is because I hadn't heard of it before.
