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

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