Matrix exponential

The exponential of a square matrix $$A$$ is the matrix denoted with $$e^{A}$$ and is by defintion:

$$e^A = I + A + \frac{A^2}{2!}+ \dots$$

The matrix exponential is defined to mirror certain properties of the regular exponential (i.e., exponential of a number), $$e^a$$. The major similarity is that the similarity of the expansions of $$e^A$$ and $$e^a$$ (see exponential).

Properties of the Matrix Exponential
Some, but not all, properties of the matrix exponential are similar to the scalar exponential.


 * If $$AB = BA$$, then $$e^{A+B}=e^A e^B$$
 * $$(e^{tA})^{-1} = e^{-tA}$$
 * $$e^{0A} = I$$
 * $$[e^{tA}]^k = e^{ktA}$$ for $$k$$ positive integer
 * $$e^{(t_2-t_1)A}e^{(t_1-t_0)A}$$ for any $$t_0, t_1, t_2$$

Solution to Linear Dynamical Systems
Another aspect of the generalization is being the solution of a linear dynamical system (or, of a first-order differential equation). Just like $$x(t) = e^{at}x(0)$$ is the solution of $$\dot{x} = a x$$, where $$a$$ is some scalar, so is $$x(t) = e^{tA} x(0)$$ the solution of the dynamical system $$\dot{x} = Ax$$, where $$A$$ is a square matrix.

In fact, $$e^{tA}$$ is the state transition matrix (denoted also as $$\Phi(t)$$), as it propagates the initial state to the state at a given time $$t$$:

$$x(t) = \Phi(t) x(0) = e^{tA} x(0)$$

In fact, it is easy to show that the transfer need not take place from the initial condition, as it can take place from a certain state $$x(\tau)$$, that is:

$$x(t+\tau) = e^{tA} x(\tau)$$

Solution to discretized LDS
The exponential matrix gives the solutions also for uniformly sampled (or discretized) versions of continuous LDSs sampled with time steps of $$h$$ (i.e., $$t_{k+1}-t_k = h$$):

$$z(k+1)=e^{hA} z(k)$$

Relation to System Stability
The matrix exponential is also used to test the stability of an LDS. Specifically, let $$\dot{x} = Ax$$ be an (autonomous) LDS. This LDS is stable if $$e^{tA}\to 0$$ as $$t \to \infty$$. The specific meaning of stability here is that $$x \to 0$$ as $$t \to \infty$$. It can be shown that this condition to stability is satisfied iff all the eigenvalue of $$A \in \mathbb{R}^{n\times n}$$ have negative real part :

$$\Re \lambda_i <0 $$, $$i=1,\dots,n$$