The matrix exponential function

What is the matrix exponential?

The matrix exponential of a square matrix is the infinite series

where is the identity matrix. This series converges for any square matrix .

How do I use the matrix exponential to solve a linear system?

The solution to the set of linear ODEs

with the initial condition

is

How do we know that this solution is correct? First, let’s check that this solution satisfies the ODEs:

Apparently, it does. Second, let’s check that this solution satisfies the initial condition:

Again, it does. (We might wonder if this is the only solution to the original ODEs --- it is, although a proof would require more work.)

How do I use the matrix exponential to solve state space models?

Consider the state space model

This model does not look the same as

Indeed, without specifying , we cannot solve for as a function of time. However, particular choices of allow us to simplify the state space model. For example, if we choose , then we can write

and so we are right back at a linear system that can be solved with the matrix exponential. Another common choice of is

for some constant matrix . (What would the size of have to be for us to define in this way?) This choice of is called state feedback, since the input depends on the state. If we plug this choice into our state space model, then we can write

and so --- just like before --- we are right back at a linear system that can be solved with the matrix exponential. Although this result will get us a long way, we will see how to solve state space models for other choices of input later on.