General Fundamental Matrix

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

Then:
 * $\Phi \left({t}\right) C$

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

Proof
$\Phi \left({t}\right) C$ is a fundamental matrix as follows:


 * $\dfrac {\mathrm d}{\mathrm d t} \Phi \left({t}\right) C = \Phi' \left({t}\right) C = A \left({t}\right) \Phi \left({t}\right) C$


 * $\det \left({\Phi \left({t}\right) C}\right) = \det \left({\Phi \left({t}\right)}\right) \det \left({C}\right) \ne 0$

Let $\Psi \left({t}\right)$ be an arbitrary fundamental matrix.

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

Hence by uniqueness:
 * $\Phi \left({t}\right) \Phi^{-1} \left({t_0}\right) \Psi \left({t_0}\right)$ is equal to $\Psi \left({t}\right)$

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