Matrix Exponential

Matrix Exponential

The matrix exponential plays an important role in solving system of linear differential equations. On this page, we will define such an object and show its most important properties. The natural way of defining the exponential of a matrix is to go back to the exponential function e^x and find a definition which is easy to extend to matrices. Indeed, we know that the Taylor polynomials
 

converges pointwise to e^x and uniformly whenever x is bounded. These algebraic polynomials may help us in defining the exponential of a matrix. Indeed, consider a square matrix A and define the sequence of matrices
 

When n gets large, this sequence of matrices get closer and closer to a certain matrix. This is not easy to show; it relies on the conclusion on e^x above. We write this limit matrix as e^A. This notation is natural due to the properties of this matrix. Thus we have the formula
 

One may also write this in series notation as
 

At this point, the reader may feel a little lost about the definition above. To make this stuff clearer, let us discuss an easy case: diagonal matrices.

Example. Consider the diagonal matrix
 

It is easy to check that
 

for . Hence we have
 

Using the above properties of the exponential function, we deduce that
 

Indeed, for a diagonal matrix A, e^A can always be obtained by replacing the entries of A (on the diagonal) by their exponentials. Now let B be a matrix similar to A. As explained before, then there exists an invertible matrix P such that
 

Moreover, we have
 

for , which implies
 

This clearly implies that
 

In fact, we have a more general conclusion. Indeed, let A and B be two square matrices. Assume that . Then we have . Moreover, if B = P^-1AP, then
 

Example. Consider the matrix
 

This matrix is upper-triangular. Note that all the entries on the diagonal are 0. These types of matrices have a nice property. Let us discuss this for this example. First, note that
 

In this case, we have
 

In general, let A be a square upper-triangular matrix of order n. Assume that all its entries on the diagonal are equal to 0. Then we have
 

Such matrix is called a nilpotent matrix. In this case, we have
 

As we said before, the reasons for using the exponential notation for matrices reside in the following properties:

Theorem. The following properties hold:

1.

;

2.

if A and B commute, meaning AB = BA, then we have
 

e^A+B = e^Ae^B;

 

 

3.

for any matrix A, e^A is invertible and