rate equations are often in this form:
where is the population vector, and is the relaxation matrix.
we can diagonalized the matrix R by solving the eigen system of R
where is the row vector formed by eigen vectors. and is diagonal matrix, such that is the eigen value for eigen vector .
by this, we can rewrite the equations as:
To find the equilibrium population, it means to solve
which mean, find the null space of R.