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.|
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 An, 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 An+1 in terms of the first n - 1 powers of A. In fact, we can repeat that process so that Am, for any arbitrary high power of m can be expressed as a linear combination of the first 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.
The inverse of a matrix exponential is given by:
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
If we have a sum of matrices in the exponent, we cannot separate them:
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.
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.