Principle Fundemental Solution Matrix and the criterion for convergence of the Matrix Expoential with function coefficients.

Question

Lately in a applied mathematic course we have been asked to solve linear ODEs via matrix solutions. In particular we defined the concept of a Principle Fundamental Solution Matrix as follows (I'm using overlines to denote vectors): $$ \text{Let} \ \overline{Y'(t)} = A(t)\overline{Y(t)} + \overline{X(t)} $$ be a system of linear ODE. Then a Principle Fundemental Solution Matrix of the above equation is a matrix $ \Phi $ such that:

1. $\overline{Y} = \Phi$ solves the above equation.

2. The initial value problem is such that $\Phi(0) = I$ where $I$ is the identity matrix.

3. If $\Phi_0$ satisfies 1 and but not 2, then $\Phi_0$ is called a Fundemental Solution Matrix. When the matrix $A$ has elements in $\mathbb{K}^N$ then such a solution matrix could be given by $e^{At}$.

As I currently understand the necessity for us to have $\Phi$ is such that, when the matrix $A(t)$ contain elements as functions, then it is not necessarily the case that the matrix exponential $e^{At}$ converges uniformly and we therefore cannot take the derivative of the sequence defining $e^{At}$ for each element in the sequence (Hence $e^{At}$ doesn't solve the system of ODE).

Since the course I'm taking is one in applied mathematic I think the reasoning given was not very convincing (But since I suspect the 'real' proof has to do areas like Functional Analysis I wouldn't think it is strange that they do so). It follows that my question concerns whether or not my understanding to the above is accurate, and that if this condition is a necessary condition (that must be satisfied )or just a sufficient condition (one of many). Moreover, I would be very glad if anyone could kindly post some references to the related theory (articles, etc), or explain some analytical intuition behind the question.

Please feel free to use common language in mathematical analysis if needed.

Thanks in advance!

Answer 1

The fundamental matrix solves the homogeneous equation $\Phi'=A(t)\,\Phi$. When the matrix $A$ depends on $t$, $\Phi(t)\ne e^{A(t)t}$. The existence of the fundamental matrix is a consequence of the existence and uniqueness theory for the linear equation. Except in the constant coefficient case, it is quite difficult (if at all possible) to have an explicit expression for the fundamental matrix.