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