I have a matrix
$ M= \begin{pmatrix} -a_1 & b_{21} & b_{31} \\ b_{12} & -a_2 & b_{32} \\ b_{13} & b_{23} & -a_3 \end{pmatrix} $
And a function $f$ which gives $ f(M)= -\frac{a_{1}^2}{9}+\frac{a_{1} a_{2}}{9}+\frac{a_{1} a_{3}}{9}-\frac{a_{2}^2}{9}+\frac{a_{2} a_{3}}{9}-\frac{a_{3}^2}{9}-\frac{b_{12} b_{21}}{3}-\frac{b_{13} b_{31}}{3}-\frac{b_{23} b_{32}}{3} $
Any thoughts on what function $f$ might represent?
I've noticed that either the covariance or second central moment can be used to represent the $a$ terms, but haven't found anything that represents the entire function or just the $b$ terms.
Thanks!
Update:
The matrix $M$ is the Jacobian of the system of differential equations given by: $ \begin{alignedat}{1} f_1 &= b_{21}n_2 + b_{31}n_3 - a_1 n_1 \\ f_2 &= b_{32}n_3 + b_{12}n_1 - a_2 n_2 \\ f_3 &= b_{13}n_1 + b_{23}n_2 - a_3 n_3 \end{alignedat} $
$f(M)$ appears while trying to solve the cubic formula for the eigenvalues of the matrix. Simpler systems have been expressable in terms of the covariance, variance, and means of $a$ and $b$.
For instance, the eigenvalues of the system given by
$ \begin{alignedat}{1} f_1&=b_{21}n_2-a_1 n_1 \\ f_2&=b_{12}n_1-a_2 n_2 \end{alignedat} $
are expressible by
$ \begin{alignedat}{1} \lambda=\pm\sqrt{\mu^2(b)-\sigma^2(b)+\sigma^2(a)}-\mu(a) \end{alignedat} $
whether the eigenvalues of the three-equation system are similarly reducible is the question.