## A Relation Between Fundamental Matrices for a First Order Linear Homogeneous System

If a real homogeneous system of first order equations has a fundamental
$n \times n$
matrix
$A$
and
$C$
is an invertible
$n \times n$
matrix, then
$B=AC$
is also a fundamental matrix.
In fact, if
$A, \: B$
are two fundamental matrices of a a system
$\mathbf{x'}=M \mathbf{x}$
then there is an invertiblem matrix
$C$
such that
$A=BC$
.
Consider the system:
$\dot{x}=y$

$\dot{y}=x$

We can write this in matrix form as
$\begin{pmatrix}\dot{x}\\ \dot{y} \end{pmatrix} = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right) \begin{pmatrix}x\\ y \end{pmatrix}$

The eigenvalues of the matrix are
$\lambda = -1, \: 1$
with corresponding eigenvectors
$\begin{pmatrix}1\\-1\end{pmatrix}, \: \begin{pmatrix}1\\1\end{pmatrix}$

A fundamental matrix is
$A= \left( \begin{array}{cc} e^t & -e^t \\ e^{-t} & e^{-t} \end{array} \right)$
.
Another fundamental matrix is
$B= \left( \begin{array}{cc} cosh t & sinh t \\ sinh t & cosh t t \end{array} \right)$
.
Then
$A=BC$
becomes
$\left( \begin{array}{cc} e^t & -e^t \\ e^{-t} & e^{-t} \end{array} \right)=\left( \begin{array}{cc} cosh t & sinh t \\ sinh t & cosh t \end{array} \right) C$
.
Hence
\begin{aligned} C &= {\left( \begin{array}{cc} cosh t & -sinh t \\ sinh t & cosh t \end{array} \right)}^{-1} \left( \begin{array}{cc} e^t & -e^t \\ e^{-t} & e^{-t} \end{array} \right) \\ &= {\left( \begin{array}{cc} cosh t & -sinh t \\ -sinh t & cosh t \end{array} \right)}^{-1} \left( \begin{array}{cc} e^t & -e^t \\ e^{-t} & e^{-t} \end{array} \right) \\ &= \left( \begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array} \right) \end{aligned}
.