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 
\[\begin{equation} \begin{aligned} C &amp;= {\left( \begin{array}{cc} cosh t &amp; -sinh t \\ sinh t &amp; cosh t \end{array} \right)}^{-1} \left( \begin{array}{cc} e^t &amp; -e^t \\ e^{-t} &amp; e^{-t} \end{array} \right) \\ &amp;= {\left( \begin{array}{cc} cosh t &amp; -sinh t \\ -sinh t &amp; cosh t \end{array} \right)}^{-1} \left( \begin{array}{cc} e^t &amp; -e^t \\ e^{-t} &amp; e^{-t} \end{array} \right) \\ &amp;= \left( \begin{array}{cc} 1 &amp; 1 \\ 1 &amp; -1 \end{array} \right) \end{aligned} \end{equation}\]
.