The State Transition Matrix to a Lin. Homo. Sys. of First Order ODEs

# The State Transition Matrix to a Linear Homogeneous System of First Order ODEs

We have already defined what a fundamental matrix to a linear homogeneous system of first order ODEs $\mathbf{x}' = A(t)\mathbf{x}$ is. We noted that these fundamental matrices are not unique. We can construct a fundamental matrix for this system from any fundamental set of solutions to this system.

We now single out a special fundamental matrix known as the state transition matrix.

 Definition: The State Transition Matrix to the linear homogeneous system of first order ODEs $\mathbf{x}' = A(t)\mathbf{x}$ on $J = (a, b)$ with $\tau \in J$ is the fundamental matrix $\Phi = \begin{bmatrix} \phi^{[1]} & \phi^{[2]} & \cdots & \phi^{[n]} \end{bmatrix}$ with $\phi^{[i]} (\tau) = e_i$ for each $i \in \{1, 2, ..., n \}$.

Here, the notation $e_i$ denotes the standard $n \times 1$ basis vector in $\mathbb{R}^n$ whose components are $0$ everywhere except for the $i^{\mathrm{th}}$ component which has the value $1$.

 Proposition 1: If $\Phi (t)$ 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 $\Phi (\tau) = I$.
• Proof: Since $\phi^{[i]}(\tau) = e_i$ for each $i \in \{ 1, 2, ..., n \}$ we have that:
(1)
\begin{align} \quad \Phi (\tau) &= \begin{bmatrix} \phi^{[1]}(\tau) & \phi^{[2]}(\tau) & \cdots & \phi^{[n]}(\tau) \end{bmatrix} \\ &= \begin{bmatrix} e_1 & e_2 & \cdots & e_n \end{bmatrix} \\ &= \begin{bmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{bmatrix} \\ &= I \end{align}
 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 $J = (a, b)$ with $\tau \in J$, then for any fundamental matrix $\Psi$ to this system on $J$ we have that $\Phi(t) = \Phi (t, \tau) = \Psi (t) \Psi^{-1} (\tau)$.
• Proof: Let $\Psi$ be any fundamental matrix of $\mathbf{x}' = A(t)\mathbf{x}$. Then for $\tau \in J$ we have that:
(2)
\begin{align} \quad \Phi (\tau) = \Phi (\tau, \tau) = \Psi (\tau) \Psi^{-1} (\tau) = I \end{align}
• So $\phi^{[i]}(\tau) = e_i$ for each $i \in \{ 1, 2, ..., n \}$, and so indeed, this formula is valid for the state transition matrix of $\mathbf{x}' = A(t)\mathbf{x}$ on $J$. $\blacksquare$

In the above proposition we used the notation $\Phi (t, \tau)$ to emphasize the dependence on both $t$ and $\tau$. When unambiguous we will use the simpler notation $\Phi (t)$.