Matrix Exponential (As a Fundamental Matrix)

From Problem Set 10: Exercise 1.1

Considering a general system of 1st order ODEs, we set up the system and environment to give us

, where A is some n x n coefficient matrix with the initial data of .

To consider the fundamental exponential matrix, we get a solution that has the form/looks like

The fundamental matrix is given to us as the solution, as a fundamental set that solves . The P(t) gives us the solution (coefficient only matrix).

The fundamental matrix X(t) given as a matrix with the columns as vectors where the columns are linearly independent.

The solutions in terms of only the fundamental matrix look like , where the C term is the vectors given by the constants c1,c2,c3...

Manipulating the system for the initial condition of some t (such that we have sufficient information to solve for c), we get .

Combining all of these, we get the general solution as some function of the fundamental matrix.

. The is the term and is known as the matrix exponential.

We see the similarity to the x' = Ax solution. We also know that , where I is the identity matrix for the given order.

This is because the matrix and inverse cancel each other out.

In the exponential form, we see that the looks like the term = . Its differential gives us the , since it is a Taylor expansion. This satisfies the 0 condition of the IVP.