## Monday, April 19, 2010

### M Theory Lesson 315

For $\omega$ the cubed root of unity, the matrix $\textrm{exp}(iM)$ is the circulant given by the three complex numbers
$0.336669 + 0.047040 i$
$0.372403 - 0.597981 i$
$0.290928 + 0.550941 i$
and one readily checks that the real parts add to $1$ while the imaginary parts add to zero. The first number is independent of the angle $\omega$ that defines $M$. For example, for $\omega = 2/9$ the second and third real parts become $-0.63665$ and $1.29998$ respectively. Note that the sum of three squares also equals $1$, making $\textrm{exp}(iM)$ doubly magic.