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

When

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

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*^{-1}*AP*, 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

Such matrix is called a

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*^{A}*e*^{B};

**3.**- for any matrix
*A*,*e*^{A}is invertible and

**
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**Author**: M.A. Khamsi

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