Fund. Mat. to a Linear Homogeneous System of First Order ODEs

# Fundamental Matrices to a Linear Homogeneous System of First Order ODEs

Recall from the Fundamental Sets of Solutions to a Linear Homogeneous System of First Order ODEs page that a fundamental set of solutions to the linear homogeneous system $\mathbf{x} = A(t) \mathbf{x}$ on $J = (a, b)$ is a set of solutions $\{ \phi^{[1]}, \phi^{[2]}, ..., \phi^{[n]} \}$ on $J$ that are linearly independent.

We now define what is called a Fundamental matrix to a linear homogeneous system of first order ODEs.

 Definition: If $\{ \phi^{[1]}, \phi^{[2]}, ..., \phi^{[n]} \}$ is a fundamental set of solutions to the linear homogeneous system of $n$ first order ODEs $\mathbf{x}' = A(t) \mathbf{x}$ on $J = (a, b)$ then the corresponding Fundamental Matrix is $\Phi = \begin{bmatrix} \phi^{[1]} & \phi^{[2]} & \cdots & \phi^{[n]} \end{bmatrix}$.

Note that $\Phi$ is an $n \times n$ matrix and can be written explicitly as:

(1)
\begin{align} \quad \Phi = \begin{bmatrix} \phi^{[1]} & \phi^{[2]} & \cdots & \phi^{[n]} \end{bmatrix} = \begin{bmatrix} \phi_1^{[1]} & \phi_1^{[2]} & \cdots & \phi_1^{[n]} \\ \phi_2^{[1]} & \phi_2^{[2]} & \cdots & \phi_2^{[n]} \\ \vdots & \vdots & \ddots & \vdots \\ \phi_n^{[1]} & \phi_n^{[2]} & \cdots & \phi_n^{[n]} \\ \end{bmatrix} \end{align}

We have previously considered the following linear homogeneous system of $2$ first order ODEs:

(2)
\begin{align} \quad x_1' &= x_1 \\ \quad x_2' &= 2x_2 \end{align}

We found that $\phi^{[1]} = \begin{bmatrix} e^t \\ 0 \end{bmatrix}$ and $\phi^{[2]} = \begin{bmatrix} 0 \\ e^{2t} \end{bmatrix}$ are solutions to this system and that $\{ \phi^{[1]}, \phi^{[2]} \}$ is a fundamental set of solutions to this system. The corresponding $2 \times 2$ fundamental matrix is given by:

(3)
\begin{align} \quad \Phi = \begin{bmatrix} \phi^{[1]} & \phi^{[2]} \end{bmatrix} = \begin{bmatrix} e^t & 0 \\ 0 & e^{2t} \end{bmatrix} \end{align}

Note that we can obtain many different fundamental sets of solutions and hence we can obtain many different corresponding fundamental matrices to a system.

Unless otherwise stated, the content of this page is licensed under Creative Commons Attribution-ShareAlike 3.0 License