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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.



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.

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Where are the statistics? – Did Jan 10 '12 at 6:31
I'm afraid I don't follow, @DidierPiau. My expectation is that $f$ will somehow relate back to means, variances, covariances, or some other quantity generally used in statistics, as this has been the case in similar systems I've already worked with. – Richard Jan 10 '12 at 6:58
I think @Didier was making the point that you didn't tell us anything about the matrix $M$ that might link it to statistics. Is it a covariance matrix? A random matrix? A joint probability distribution? – joriki Jan 10 '12 at 7:12
The matrix $M$ may not bear a direct connection to statistics, but I have reason to believe it may be expressible in terms of statistical functions. I've updated the question accordingly. @DidierPiau – Richard Jan 10 '12 at 19:52
up vote 3 down vote accepted

This is $$\frac1{18}(\operatorname{Tr}M)^2-\frac16\operatorname{Tr}M^2\;.$$

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Any thoughts on whether this might relate to some more commonly used function? I don't recognise this one off-handedly. – Richard Jan 10 '12 at 6:59
It reminds me of the character of the antisymmetric tensor product of a group representation, which is $((\operatorname{Tr}M)^2-\operatorname{Tr}M^2)/2$, but it's not exactly that and I don't know of any connection to statistics. It might help if you tell us what $M$ is. – joriki Jan 10 '12 at 7:28
I've done so, please see above, @joriki. – Richard Jan 10 '12 at 19:53

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