# Engineering Analysis/Matrix Exponentials

Before reading this chapter, students should know what the taylor series of a function is, and how to obtain it. This is discussed in Calculus. |

## Matrix Exponentials[edit | edit source]

If we have a matrix *A*, we can raise that matrix to a power of *e* as follows:

It is important to note that this is not necessarily (not usually) equal to each individual element of *A* being raised to a power of *e*. Using taylor-series expansion of exponentials, we can show that:

- .

In other words, the matrix exponential can be reduced to a sum of powers of the matrix. This follows from both the taylor series expansion of the exponential function, and the cayley-hamilton theorem discussed previously.

However, this infinite sum is expensive to compute, and because the sequence is infinite, there is no good cut-off point where we can stop computing terms and call the answer a "good approximation". To alleviate this point, we can turn to the Cayley-Hamilton Theorem. Solving the Theorem for *A ^{n}*, we get:

Multiplying both sides of the equation by *A*, we get:

We can substitute the first equation into the second equation, and the result will be *A ^{n+1}* in terms of the first

*n - 1*powers of

*A*. In fact, we can repeat that process so that

*A*, for any arbitrary high power of m can be expressed as a linear combination of the first

^{m}*n - 1*powers of

*A*. Applying this result to our exponential problem:

Where we can solve for the α terms, and have a finite polynomial that expresses the exponential.

## Inverse[edit | edit source]

The inverse of a matrix exponential is given by:

## Derivative[edit | edit source]

The derivative of a matrix exponential is:

Notice that the exponential matrix is commutative with the matrix *A*. This is not the case with other functions, necessarily.

## Sum of Matrices[edit | edit source]

If we have a sum of matrices in the exponent, we cannot separate them:

## Differential Equations[edit | edit source]

If we have a first-degree differential equation of the following form:

With initial conditions

Then the solution to that equation is given in terms of the matrix exponential:

This equation shows up frequently in control engineering.

## Laplace Transform[edit | edit source]

As a matter of some interest, we will show the Laplace Transform of a matrix exponential function:

We will not use this result any further in this book, although other books on engineering might make use of it.