Engineering Analysis/Matrix Exponentials

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Engineering Analysis
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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.

Contents

[edit] Matrix Exponentials

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

eA

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:

e^{A} = I + A + \frac{1}{2}A^2 + \frac{1}{6}A^3 + ... = \sum_{k=0}^\infty{1 \over k!}A^k.

In other words, the matrix exponential can be reducted 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:

A^n = -c_{n-1}A^{n-1} - c_{n-2}A^{n-2} - \cdots - c_1A - c_0I

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

A^{n+1} = -c_{n-1}A^n - c_{n-2}A^{n-1} - \cdots - c_1A^2 - c_0A

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:

e^A = \alpha_0I + \alpha_1A + \cdots + \alpha_{n-1}A^{n-1}

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

[edit] Inverse

The inverse of a matrix exponential is given by:

(eA) − 1 = e − A

[edit] Derivative

The derivative of a matrix exponential is:

\frac{d}{dx}e^{Ax} = Ae^{Ax} = e^{Ax}A

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

[edit] Sum of Matrices

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

e^{(A+B)x} \ne e^{Ax}e^{Bx}

[edit] Differential Equations

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

x'(t) = Ax(t) + f(x)

With initial conditions

x(t0) = c

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

x(t) = e^{a(t - t_0)}c + \int_{t_0}^t e^{a(t - \tau)}f(\tau)d\tau

This equation shows up frequently in control engineering.

[edit] Laplace Transform

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

\mathcal{L}[e^{At}] = (sI - A)^{-1}

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

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