The
Koide mass matrices are written in terms of two parameters, which we will call $n$ (placed on the diagonal) and $\phi$ (the phase angle). The eigenvalues of such a matrix may be expressed as $\lambda + n$ for an eigenvalue $\lambda$ of
since adding $n \mathbf{1}$ to the Koide matrix leaves the eigenvectors intact. Note that the eigenvalue set for $K$ always sums to zero, so these three $\lambda$ describe the mass differences within a triplet. For example, at $\phi = 2/9$ the eigenvalues of $K$ are $\{ -0.59367, -1.35715, 1.9508 \}$. At a cubed root, the eigenvalues are $\{ -1, -1, 2 \}$, like the entries of the
idempotent $S$. All possible zero mass triplets are thus described by the $U(1)$ of the variable $\phi$. As $\phi$ varies, a given eigenvalue varies between $-2$ and $+2$.
since adding $n \mathbf{1}$ to the Koide matrix leaves the eigenvectors intact. Note that the eigenvalue set for $K$ always sums to zero, so these three $\lambda$ describe the mass differences within a triplet. For example, at $\phi = 2/9$ the eigenvalues of $K$ are $\{ -0.59367, -1.35715, 1.9508 \}$. At a cubed root, the eigenvalues are $\{ -1, -1, 2 \}$, like the entries of the idempotent $S$. All possible zero mass triplets are thus described by the $U(1)$ of the variable $\phi$. As $\phi$ varies, a given eigenvalue varies between $-2$ and $+2$. 
Wow, Carl, MAXIMA is cool!
ReplyDeleteWith the graphics, it appears you're getting more out of it than I am. I use it for algebra only. Why don't you publish the Maxima input?
ReplyDeleteCarl
When I find something more interesting than basic mixtures of sines and cosines, then I'll explain what the functions are. But note that this was incredibly easy to figure out how to use, with the help menu and a little experimentation.
ReplyDelete