Example 1. Consider
the matrix ,
1 (a) Find .
1 (b) Find .
Solution 1 (b).
We want to find
First look at some powers
Now use the calculation
Find the expression for the general term
Find matrix exponential will
be the sum of the infinite series
The sum of the first five terms is
The sum of the first ten terms is
Each element in can be calculated by the sum of an infinite series and Mathematica can assist us in these computations.
Therefore, the matrix exponential is
This can be compared to the matrix exponential that can be computed by using Mathematica's built in procedure MatrixExp[At].
Caveat. This shows the power of "artificial intelligence" that is available in Mathematica. For this example the matrix had a full set of eigenvectors. Mathematica is "smart" enough to know how to compute the matrix exponential for the difficult case when the set of eigenvectors is deficient.
(c) John H. Mathews 2004