rate equations are often in this form:

$\frac{dN}{dt} = R\cdot N$

where $N={N_1,N_2,...,N_n}$ is the population vector, and $R$ is the relaxation matrix.

we can diagonalized the matrix R by solving the eigen system of R

$R = E\cdot D \cdot E^{-1}$

where $E = {v_1, v_2, ... v_n}$ is the row vector formed by eigen vectors. and $D$ is diagonal matrix, such that $D_{ii} = \lambda_i$ is the eigen value for eigen vector $v_i$.

by this, we can rewrite the equations as:

$\frac{dN}{dt} = E \cdot D \cdot E^{-1} \cdot N$

$E^{-1} \cdot\frac{dN}{dt} = D \cdot (E^{-1}\cdot N)$

$\frac{ d (E^{-1}\cdot N) }{dt} = D \cdot (E^{-1}\cdot N)$

$\frac{dV}{dt} = D\cdot V$

$\frac{dV_i}{dt}=D_{ii} V_i =\lambda_i V_i$

$V_i = V_i(0) Exp(\lambda_i t)$

$N_i = E_{ij} V_j$

To find the equilibrium population, it means to solve

$R\cdot N = 0$

which mean, find the null space of R.

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