Matrix exponentials are important in the solution of systems of ordinary differential equations (e.g., Bellman 1970).

In some cases, it is a simple matter to express the matrix exponential. For example, when
is a diagonal matrix, exponentiation can be performed
simply by exponentiating each of the diagonal elements. For example, given a diagonal
matrix

(5)

The matrix exponential is given by

(6)

Since most matrices are diagonalizable,
it is easiest to diagonalize the matrix before exponentiating it.

Bellman, R. E. Introduction to Matrix Analysis, 2nd ed. New York: McGraw-Hill, 1970.Horn,
R. A. and Johnson, C. R. Topics
in Matrix Analysis. Cambridge, England: Cambridge University Press, p. 208,
1994.Moler, C. and van Loan, C. "Nineteen Dubious Ways to Compute
the Exponential of a Matrix, Twenty-Five Years Later." SIAM Rev.45,
3-49, 2003.