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Suppose that the functions $a_{ij}(t), 1\leq i,j \leq n$ are continuous on $|t-a|\leq T$. Let $\Phi (t)$ and $\Psi (t)$ be two fundamental matrices of the homogeneous system

$$\frac{d\mathbf{x}}{dt}=\mathbf{A}(t)\mathbf{x}, \ \ \mathbf{A}(t)=(a_{ij}(t))$$.

Prove that $\Phi(t)\Psi^{-1}(t)$ is a constant matrix.

My trial: Since we have the general solution $\mathbf{x}(t)=\Phi (t)\mathbf{c}= \Psi(t)\mathbf{c}$, hecne $\mathbf{x} (t) = \Phi (t) \Psi^{-1}(t) \mathbf{x}(t)$ is another solution to the system.

But then I dont know how to proceed, thanks in advance.

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1 Answer 1

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Using the two fundamental matrices of the homogeneous system, we can assert that $\mathbf{x}(t)=\mathbf{\Phi}(t)\mathbf{I}$ is one of its solutions and there must exist some constant matrix $\mathbf{C}$ satisfying $$\mathbf{x}(t)=\mathbf{\Phi}(t)\mathbf{I}=\mathbf{\Psi}(t)\mathbf{C}$$, where $\mathbf{I}$ is the identity matrix. If $\mathbf{\Psi}(t)$ is reversible, then the formula above can be changes as $$\mathbf{\Phi}(t)\mathbf{I}\mathbf{\Psi}^{-1}(t)=\mathbf{\Phi}(t)\mathbf{\Psi}^{-1}(t)=\mathbf{C}$$, i.e., $\mathbf{\Phi}(t)\mathbf{\Psi}^{-1}(t)$ is a constant matrix.

Hope this can help you.

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