The Matrix Exponential as a Limit of Powers 2.6 2002/06/25 2002/07/10 00:00:00.006 GMT-5 Steven Cox cox@rice.edu Liqun Wang liqun@rice.edu Steven Cox cox@rice.edu matrix exponential limit power This module describes how to compute the matrix exponential using a limit of powers. You may recall from Calculus that for any numbers a and t one may achieve a t via a t k 1 a t k k The natural matrix definition is therefore A t k I A t k k where I is the n-by-n identity matrix. The easiest case is the diagonal case, e.g., A 1 0 0 2 for then I A t k k 1 t k k 0 0 1 2 t k k and so (recalling above) A t t 0 0 2 t Note that this is NOT the exponential of each element of A. As a concrete example let us suppose A 0 1 -1 0 From I A t 1 t t 1 I A t 2 2 1 t 2 t 2 1 1 t 2 t 2 1 1 t 2 4 t t 1 t 2 4 I A t 3 3 1 t 2 3 t t 3 27 t t 3 27 1 t 2 3 I A t 4 4 -3 t 2 8 t 4 256 1 t t 3 16 t t 3 16 -3 t 2 8 t 4 256 1 I A t 5 5 -2 t 2 5 t 4 125 1 t 2 t 3 25 t 5 3125 t 2 t 3 25 t 5 3125 -2 t 2 5 t 4 125 1 We discern a pattern: the diagonal elements are equal even polynomials while the off diagonal elements are equal but opposite odd polynomials. The degree of the polynomial will grow with k and in the limit we 'recognize' A t t t t t If A 0 1 0 0 then I A t k k 1 t 0 1 for each value of k and so A t 1 t 0 1