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.<br /> 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
$A=BC$
. <br /> Consider the system:<br />
$\dot{x}=y$
<br />
$\dot{y}=x$
<br /> We can write this in matrix form as
$\begin{pmatrix}\dot{x}\\ \dot{y} \end{pmatrix} = \left( \begin{array}{cc} 0 &amp; 1 \\ 1 &amp; 0 \end{array} \right) \begin{pmatrix}x\\ y \end{pmatrix}$
<br /> The eigenvalues of the matrix are
$\lambda = -1, \: 1$
with corresponding eigenvectors
$\begin{pmatrix}1\\-1\end{pmatrix}, \: \begin{pmatrix}1\\1\end{pmatrix}$
<br /> A fundamental matrix is
$A= \left( \begin{array}{cc} e^t &amp; -e^t \\ e^{-t} &amp; e^{-t} \end{array} \right)$
.<br /> Another fundamental matrix is
$B= \left( \begin{array}{cc} cosh t &amp; sinh t \\ sinh t &amp; cosh t t \end{array} \right)$
.<br /> Then
$A=BC$
becomes<br />
$\left( \begin{array}{cc} e^t &amp; -e^t \\ e^{-t} &amp; e^{-t} \end{array} \right)=\left( \begin{array}{cc} cosh t &amp; sinh t \\ sinh t &amp; cosh t \end{array} \right) C$
.<br /> Hence
\begin{aligned} C &amp;= {\left( \begin{array}{cc} cosh t &amp; -sinh t \\ sinh t &amp; cosh t \end{array} \right)}^{-1} \left( \begin{array}{cc} e^t &amp; -e^t \\ e^{-t} &amp; e^{-t} \end{array} \right) \\ &amp;= {\left( \begin{array}{cc} cosh t &amp; -sinh t \\ -sinh t &amp; cosh t \end{array} \right)}^{-1} \left( \begin{array}{cc} e^t &amp; -e^t \\ e^{-t} &amp; e^{-t} \end{array} \right) \\ &amp;= \left( \begin{array}{cc} 1 &amp; 1 \\ 1 &amp; -1 \end{array} \right) \end{aligned}
.