Abel's Fundamental Matrix Formula
Recall from the Fundamental Matrices to a Linear Homogeneous System of First Order ODEs that if $\{ \phi^{[1]}, \phi^{[2]}, ..., \phi^{[n]} \}$ is a fundamental set of solutions to a linear homogeneous system of $n$ first order ODEs $\mathbf{x}' = A(t) \mathbf{x}$ then the corresponding fundamental matrix is defined as the $n \times n$ matrix:
(1)We will now prove an important result known as Abel's Fundamental Matrix formula.
Theorem 1 (Abel's Fundamental Matrix Formula): If $\Phi$ is a fundamental matrix to the linear homogeneous system $\mathbf{x}' = A(t)\mathbf{x}$ then $\Phi$ is a solution to the matrix equation $X' = A(t)X$. |
We define $X' = [x_{i,j}']$ where $X = [x_{i,j}]$.
- Proof: Let $\Phi$ be a fundamental matrix to the linear homogeneous system $\mathbf{x}' = A(t)\mathbf{x}$. Then each of the columns of $\Phi$ is a solution to $\mathbf{x}' = A(t)\mathbf{x}$ on some predescribed interval $J = (a, b)$. So:
- Hence $\Phi$ is a solution to the matrix equation $X' = A(t)X$. $\blacksquare$
We have already looked at the following linear homogeneous system of $2$ first order ODEs:
(3)We have that $A(t) = \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}$.
We have found a fundamental matrix for the system:
(4)We will show that $\Phi$ is the solution to the matrix equation $X' = A(t) X$, i.e., $\Phi' = A(t)\Phi$. We first compute the lefthand side of this equation:
(5)And now the righthand side of this equation:
(6)We see that indeed $\Phi' = A(t)\Phi$, i.e., $\Phi$ is a solution to the matrix equation $X' = A(t)X$.