the principal fundamental matrix
A fundamental matrix for the system x' = A x is a square matrix whose columns are a full set of independent solution vectors, packed side by side. Any such matrix solves the matrix equation X' = A X and lets you write every solution as X(t) c for some constant column c. But there are infinitely many fundamental matrices — you can rescale or recombine the columns however you like — and most of them carry no obvious meaning at the starting time. The principal fundamental matrix is the one special, normalized choice that makes the bookkeeping vanish.
It is the fundamental matrix that equals the identity at the chosen initial time. If we start at t = 0, the principal fundamental matrix Phi(t) satisfies Phi' = A Phi together with Phi(0) = I. The payoff is immediate: the solution with initial state x(0) is simply x(t) = Phi(t) x(0), with no constants left to solve for, because at t = 0 the matrix is I and returns x(0) untouched. If you already have any fundamental matrix X(t), you build the principal one by Phi(t) = X(t) X(0)^(-1) — multiply on the right by the inverse of its own value at the start, which renormalizes it to be I there.
For a constant-coefficient system this principal fundamental matrix is exactly the matrix exponential: Phi(t) = e^(At). That is the bridge between the abstract idea (a normalized basis of solutions) and the concrete object (the exponential series). For variable-coefficient systems x' = A(t) x there is no e^(At), but a principal fundamental matrix still exists; it is the right general notion, with the matrix exponential as its constant-coefficient special case.
Take x' = [2, 0; 0, 3] x. Two solutions are (e^(2t), 0) and (0, e^(3t)); side by side they form X(t) = [e^(2t), 0; 0, e^(3t)]. Here X(0) = I already, so X is itself the principal fundamental matrix, and the solution from x(0) = (4, 5) is just (4 e^(2t), 5 e^(3t)). If instead the columns had started scrambled, you would right-multiply by X(0)^(-1) to reset them to I at t = 0.
Normalizing a fundamental matrix to be I at the start turns 'solve for the constants' into 'read off the answer'.
The identity condition is tied to a chosen starting time: Phi(t0) = I makes Phi normalized at t0, not at every time — at other times it is generally not the identity.