Props. of the St. Trans. Mat. to a Lin. Homo. Sys. of First Order ODEs

# Basic Properties of the State Transition Matrix to a Linear Homogeneous System of First Order ODEs

Recall from The State Transition Matrix to a Linear Homogeneous System of First Order ODEs page that the state transition matrix to a linear homogeneous system of first order ODEs $\mathbf{x}' = A(t)\mathbf{x}$ on $J = (a, b)$ with $\tau \in J$ is a fundamental matrix $\Phi = \begin{bmatrix} \phi^{} & \phi^{} & \cdots & \phi^{[n]} \end{bmatrix}$ such that $\phi^{[i]}(\tau) = e_i$ for each $i \in \{ 1, 2, ..., n \}$.

We will now prove some basic properties of state transition matrices.

 Proposition 1: If $\Phi (t, \tau)$ is the state transition matrix to the linear homogeneous system of first order ODEs $\mathbf{x}' = A(t)\mathbf{x}$ on $J = (a, b)$ with $\tau \in J$ then $\Phi (t, \tau)$ is nonsingular for every $t \in J$.
• Proof: Let $\Psi$ be any fundamental matrix to $\mathbf{x}' = A(t)\mathbf{x}$ on $J$. Then:
(1)
\begin{align} \quad \Phi (t, \tau) = \Psi (t) \Psi^{-1} (\tau) \end{align}
• We take the determinant of both sides of the equation above:
(2)
\begin{align} \quad \det \Phi (t, \tau) = (\det \Psi (t)) \cdot (\det \Psi^{-1} (\tau)) \end{align}
• Since $\Psi$ is a fundamental matrix, $\det \Psi (t) \neq 0$ for all $t \in J$ and $\Psi^{-1} (\tau) \neq 0$. Therefore $\det \Phi (t, \tau) \neq 0$ for all $t \in J$. So $\Phi (t, \tau)$ is nonsingular for every $t \in J$. $\blacksquare$
 Proposition 2: If $\Phi$ is the state transition matrix to the linear homogeneous system of first order ODEs $\mathbf{x}' = A(t)\mathbf{x}$ on $J = (a, b)$ then for all $t, u, v \in J$ we have that $\Phi (t, v) = \Phi (t, u) \Phi (u, v)$.
• Proof: Let $\Psi$ be any fundamental matrix to $\mathbf{x}' = A(t)\mathbf{x}$ on $J$. Then:
(3)
\begin{align} \quad \Phi (t, u) = \Psi(t) \Psi^{-1}(u) \quad \mathrm{and} \quad \Phi (u, v) = \Psi (u) \Psi^{-1} (v) \end{align}
• Therefore:
(4)
\begin{align} \quad \Phi (t, v) = \Psi (t) \Psi^{-1}(v) = \Psi (t) I \Psi^{-1} (V) = [\Psi(t) \Psi^{-1}(u)][\Psi (u) \Psi^{-1} (v)] = \Phi(t, u) \Phi (u, v) \quad \blacksquare \end{align}
 Proposition 3: If $\Phi$ is the state transition matrix to the linear homogeneous system of first order ODEs $\mathbf{x}' = A(t)\mathbf{x}$ on $J = (a, b)$ and $\tau \in J$ then $[\Phi (t, \tau)]^{-1} = \Phi (\tau, t)$ for all $t \in J$.
• Proof: Let $\Psi$ be any fundamental matrix to $\mathbf{x}' = A(t)\mathbf{x}$ on $J$. Then:
(5)
\begin{align} \quad \Phi(t, \tau) \Phi (\tau, t) = \Phi(t, t) = I \end{align}
• Therefore:
(6)
\begin{align} \quad [\Phi (t, \tau)]^{-1} = \Phi (\tau, t) \quad \blacksquare \end{align}