M Theory Lesson 301

Some time ago we looked at a nonassociative product for matrices with ordinary commutative number entries, using exponentiation and multiplication. For $3 \times 3$ circulants this product takes the form Consider a Koide mass matrix with two real parameters, $r = e^x$ and $\theta$. This may now be expressed in the form which puts the mass gap between distinct eigenvalues into a simple complex number, $x \pm i \theta$. Note that any base may replace $e$ here, so long as logarithms are taken in that base. Scalar multiplication of the left hand matrix by $\lambda$ results in a rescaling of $\lambda^{2x}$, so for the special value of $x = 1/2$ this nonassociative product preserves scale. Scalar multiplication on the right would not be permitted.