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  • https://math.libretexts.org/Bookshelves/Linear_Algebra/Matrix_Analysis_(Cox)/05%3A_Matrix_Methods_for_Dynamical_Systems/5.02%3A_The_Laplace_Transform
    The Laplace Transform of a matrix of functions is simply the matrix of Laplace transforms of the individual elements. \[ \begin{align*} \mathscr{L}(B \textbf{x}+\textbf{g}) &= \mathscr{L}(B \textbf{x}...The Laplace Transform of a matrix of functions is simply the matrix of Laplace transforms of the individual elements. \[ \begin{align*} \mathscr{L}(B \textbf{x}+\textbf{g}) &= \mathscr{L}(B \textbf{x})+\mathscr{L}(\textbf{g}) \\[4pt] &= B \mathscr{L}(\textbf{x})+\mathscr{L}(\textbf{g}) \end{align*}\] We note that \((sI-B)^{-1}\) well defined except at the roots of the quadratic, \(s^{2}-4s+3\) determinant of \((sI-B)\) and is often referred to as the characteristic polynomial of \(B\).
  • https://math.libretexts.org/Bookshelves/Linear_Algebra/Matrix_Analysis_(Cox)/08%3A_The_Eigenvalue_Problem/8.02%3A_The_Resolvent
    \[\frac{1}{s-b} = \frac{\frac{1}{s}}{1-\frac{b}{s}} = \frac{1}{s}+\frac{b}{s^2}+ \cdots +\frac{b^{n-1}}{s^n}+\frac{b^n}{s^n} \frac{1}{s-b} \nonumber\] \[(sI-B)^{-1} = s^{-1} \left(I-\frac{B}{s}\right)...\[\frac{1}{s-b} = \frac{\frac{1}{s}}{1-\frac{b}{s}} = \frac{1}{s}+\frac{b}{s^2}+ \cdots +\frac{b^{n-1}}{s^n}+\frac{b^n}{s^n} \frac{1}{s-b} \nonumber\] \[(sI-B)^{-1} = s^{-1} \left(I-\frac{B}{s}\right)^{-1} \left(\frac{1}{s}+\frac{B}{s^2}+ \cdots +\frac{B^{n-1}}{s^n}+\frac{B^n}{s^n}(sI-B)^{-1}\right) \nonumber\] \[(s_{2}I-B)^{-1}-(s_{1}I-B)^{-1} = (s_{2}I-B)^{-1}(s_{1}I-B-s_{2}I+B)(s_{1}I-B)^{-1} \nonumber\]

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