10.1: Overview

The matrix exponential is a powerful means for representing the solution to nn linear, constant coefficient, differential equations. The initial value problem for such a system may be written

$$x′(t) = Ax(t) \nonumber$$

$$x(0) = x_{0} \nonumber$$

where $A$ is the n-by-n matrix of coefficients. By analogy to the 1-by-1 case we might expect

$$x(t) = e^{At}u \nonumber$$

to hold. Our expectations are granted if we properly define $e^{At}$. Do you see why simply exponentiating each element of $At$ will no suffice?

There are at least 4 distinct (but of course equivalent) approaches to properly defining $e^{At}$. The first two are natural analogs of the single variable case while the latter two make use of heavier matrix algebra machinery.

1. The Matrix Exponential as a Limit of Powers
2. The Matrix Exponential as a sum of Powers
3. The Matrix Exponential via the Laplace Transform
4. The Matrix Exponential via Eigenvalues and Eigenvectors

Please visit each of these modules to see the definition and a number of examples.

For a concrete application of these methods to a real dynamical system, please visit the Mass-Spring-Damper-module.

Regardless of the approach, the matrix exponential may be shown to obey the 3 lovely properties

1. $\frac{d}{dt}(e^{At}) = Ae^{At} = e^{At}A$
2. $e^{A(t_{1}+t_{2})} = e^{At_{1}}e^{At_{2}}$
3. $e^{At}$ is nonsingular and $(e^{At})^{-1} = e^{-(At)}$

Let us confirm each of these on the suite of examples used in the submodules.

Example $\PageIndex{1}$

If

$$A = \begin{pmatrix} {1}&{0}\\ {0}&{2} \end{pmatrix} \nonumber$$

then

$$e^{At} = \begin{pmatrix} {e^t}&{0}\\ {0}&{e^{2t}} \end{pmatrix} \nonumber$$

1. $\frac{d}{dt}(e^{At}) = \begin{pmatrix} {e^t}&{0}\\ {0}&{e^{2t}} \end{pmatrix} = \begin{pmatrix} {1}&{0}\\ {0}&{2} \end{pmatrix} \begin{pmatrix} {e^t}&{0}\\ {0}&{e^{2t}} \end{pmatrix}$
2. $\begin{pmatrix} {e^{t_{1}+t_{2}}}&{0}\\ {0}&{e^{2t_{1}+2t_{2}}} \end{pmatrix} = \begin{pmatrix} {e^{t_{1}}e^{t_{2}}}&{0}\\ {0}&{e^{2t_{1}}e^{2t_{2}}} \end{pmatrix} = \begin{pmatrix} {e^{t_{1}}}&{0}\\ {0}&{e^{2t_{1}}} \end{pmatrix} \begin{pmatrix} {e^{t_{2}}}&{0}\\ {0}&{e^{2t_{2}}} \end{pmatrix}$
3. $(e^{At})^{-1} = \begin{pmatrix} {e^{-t}}&{0}\\ {0}&{e^{-(2t)}} \end{pmatrix} = e^{-(At)}$

Example $\PageIndex{2}$

If

$$A = \begin{pmatrix} {0}&{1}\\ {-1}&{0} \end{pmatrix} \nonumber$$

then

$$e^{At} = \begin{pmatrix} {\cos(t)}&{\sin(t)}\\ {-\sin(t)}&{\cos(t)} \end{pmatrix} \nonumber$$

1. $\frac{d}{dt}(e^{At}) = \begin{pmatrix} {-\sin(t)}&{\cos(t)}\\ {-\cos(t)}&{-\sin(t)} \end{pmatrix}$ and $Ae^{At} = \begin{pmatrix} {-\sin(t)}&{\cos(t)}\\ {-\cos(t)}&{-\sin(t)} \end{pmatrix}$
2. You will recognize this statement as a basic trig identity $\begin{pmatrix} {\cos(t_{1}+t_{2})}&{\sin(t_{1}+t_{2})}\\ {-\sin(t_{1}+t_{2})}&{\cos(t_{1}+t_{2})} \end{pmatrix} = \begin{pmatrix} {\cos(t_{1})}&{\sin(t_{1})}\\ {-\sin(t_{1})}&{\cos(t_{1})} \end{pmatrix} \begin{pmatrix} {\cos(t_{2})}&{\sin(t_{2})}\\ {-\sin(t_{2})}&{\cos(t_{2})} \end{pmatrix}$
3. $(e^{At})^{-1} = \begin{pmatrix} {\cos(t)}&{-\sin(t)}\\ {\sin(t)}&{\cos(t)} \end{pmatrix} = \begin{pmatrix} {\cos(-t)}&{-\sin(-t)}\\ {\sin(-t)}&{\cos(-t)} \end{pmatrix} = e^{-(At)}$

Example $\PageIndex{3}$

If

$$A = \begin{pmatrix} {0}&{1}\\ {0}&{0} \end{pmatrix} \nonumber$$

then

$$e^{At} = \begin{pmatrix} {1}&{t}\\ {0}&{1} \end{pmatrix} \nonumber$$

1. $\frac{d}{dt}(e^{At}) = \begin{pmatrix} {0}&{1}\\ {0}&{0} \end{pmatrix} = Ae^{At}$
2. $\begin{pmatrix} {1}&{t_{1}+t_{2}}\\ {0}&{1} \end{pmatrix} = \begin{pmatrix} {1}&{t_{1}}\\ {0}&{1} \end{pmatrix} \begin{pmatrix} {1}&{t_{2}}\\ {0}&{1} \end{pmatrix}$
3. $\begin{pmatrix} {1}&{t}\\ {0}&{1} \end{pmatrix}^{-1} = \begin{pmatrix} {1}&{-t}\\ {0}&{1} \end{pmatrix} = e^{-At}$