For a general unimolecular reaction involving interconversion of  different species, whose concentrations at time  are denoted by  through , an analytic form for the time-evolution of the species can be found. Let the rate constant of conversion from species  to species  be denoted as , and construct a rate-constant matrix  whose entries are the .
Also, let  be the vector of concentrations as a function of time.
Let  be the vector of ones.
Let  be the  identity matrix.
Let  be the function that takes a vector and constructs a diagonal matrix whose on-diagonal entries are those of the vector.
Let  be the inverse Laplace transform from  to .
Then the time-evolved state  is given by
thus providing the relation between the initial conditions of the system and its state at time .