# Matrix representation of complex numbers in exponential form

Do there exist matrices M and P for this equation? Or perhaps M and P dont need to be matrices? I saw this and this question after googling which made me wonder about whether the exponential form of complex numbers would still work.

$z = a\begin{pmatrix} 1&0\\ 0&1 \end{pmatrix}+b\begin{pmatrix} 0&1\\ -1&0 \end{pmatrix} = \begin{pmatrix} a&b\\ -b&a \end{pmatrix} = Me^P$





I just found this article which I dont really understand, but where they solve a case similar to this. The thing I dont understand in this article is how you exponentiate with a matrix and what that alpha is doing there. Also I dont understand that in order to make this work they have to make the matrix $\Phi$ a complex matrix, looks like it defeats the purpose of having a matrix representation of complex numbers. Not sure whether to append this to my question or add as answer, because I still dont understand it.

They say, $z = \begin{pmatrix} a&jb\\ j\alpha^2b&a \end{pmatrix} = aI+b\Phi$

Where, $\Phi = \begin{pmatrix} 0&j\\ j\alpha^2&0 \end{pmatrix}$

For which they say, $e^{\phi\Phi}=cos(\alpha\phi)I+\frac{1}{\alpha}sin(\alpha\phi)\Phi$

Show that $z=\rho e^{i\theta}$ with $i\theta=\begin{pmatrix}0&\theta\\-\theta&0\end{pmatrix}$. Calculate $e^{i\theta}$ and deduce $\theta,\rho$.
• Thanks! Looks like $\rho$ and $\theta$ should be scalars then, right? Could you explain how you calculate $e^{\begin{pmatrix}0&\theta\\-\theta&0\end{pmatrix}}$? – Leo Apr 16 '14 at 13:47
• Use the definition: $e^A=I+A+A^2/2+\cdots$ ; you'll see trigonometric functions appear. – loup blanc Apr 16 '14 at 13:50
• Ah yes, it comes out nicely! Exactly nothing changes in how I calculate the magnitude and the phase as with normal notation: I ended up doing exactly the same as before. Is there a linear algebra way to calculate the "magnitude" and "phase" of $\begin{pmatrix} a&b\\ -b&a \end{pmatrix}$ ? – Leo Apr 16 '14 at 14:20
• Let $Z=\begin{pmatrix}a&b\\-b&a\end{pmatrix}$. Then $\rho=\sqrt{\det(Z)}$ and $Z=\rho O$ where $O\in O^+(2)$. Bourbaki defines an angle as an element of $O^+(2)$, identifying $\theta+2\pi\mathbb{Z}$ and $e^{i\theta}$. The effective calculation of $\theta$ is an analytic calculation. – loup blanc Apr 16 '14 at 21:09
• $O⁺(n)$ is the group of rotations in dimension $n$ (orthogonal matrices with determinant $1$). – loup blanc Apr 17 '14 at 12:17