General Fundamental Matrix

Theorem
Let $\Phi(t)$ be a fundamental matrix of the system $x' = A(t)x$.

Then
 * $\Phi(t)C$

is a general fundamental matrix of $x' = A(t)x$, where $C$ is any nonsingular matrix.

Proof
Clearly $\Phi(t)C$ is a fundamental matrix:


 * $\dfrac{d}{dt} \Phi(t)C = \Phi'(t)C = A(t)\Phi(t)C$


 * $\det( \Phi(t) C ) = \det(\Phi(t)) \det(C) \ne 0$

Let $\Psi(t)$ be an arbitrary fundamental matrix.

Then from General Vector Solution of Fundamental Matrix $\Phi(t) \Phi^{-1}(t_0) \Psi(t_0)$ solves the same matrix equation and has the same initial conditions.

Hence by uniqueness $\Phi(t) \Phi^{-1}(t_0) \Psi(t_0)$ is equal to $\Psi(t)$.

Letting $C = \Phi^{-1}(t_0) \Psi(t_0)$ finishes the proof.