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For the jacobian matrix differentail equation:

$\frac{dx}{dt}=Bx$

where B is a symmetric matrix (jacobian matrix), x is a vector.

I am interested in $x_A$ which is the average value of each element of the vector $x$.

I wonder if there is any way to get an approximate expression for $x_A$ as a function of $t$ using the eigenvalue, eigenvector or any other properties related to the jacobian matrix $B$.

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5 
 I find your wondering confusing. You can get more than an approximate expression, you can get an exact one. Up to an isometry, your solutions all look like $(e^{a_1t}x_1, \cdots, e^{a_nt}x_n)$ where $a_1, \cdots, a_n$ are the eigenvalues of $B$. So $x$ as a function of time is just the orthogonal projection of $(e^{a_1t}x_1, \cdots, e^{a_nt}x_n)$ onto some line in $\mathbb R^n$ (the line depends on the eigenvectors of $B$). – Ryan Budney Feb 26 2012 at 14:01
But maybe someone can provide a reference for Leo to look at? – Deane Yang Feb 26 2012 at 17:54
2 
 Smale and Hirsh's "Differential Equations, Dynamical Systems and Linear Algebra" is a classic for this kind of material. – Ryan Budney Feb 26 2012 at 19:49

closed as too localized by Ryan Budney, mathphysicist, Chris Godsil, Robert Israel, S. Sra Feb 26 2012 at 20:25

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