Diagonal Form

We use the idea of Eigenvectors and Eigenvalues, which gives us information about the internal dynamics of an LTI system.

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Transformation Matrix - $T=Q^{-1}$ → $\bar{x}=Q^{-1}x$

Where $Q$ is the eigenvector matrix corresponding to the eigenvalue matrix $\Lambda$

$$\begin{array}{rl}\Lambda & =\left[\begin{array}{cccc}{\lambda }_1& 0& \dots & 0\\ 0& {\lambda }_2& \dots & 0\\ ⋮& & & ⋮\\ 0& 0& 0& {\lambda }_n\end{array}\right]\\ & \\ & \\ Q& =\left[\begin{array}{cccc}{q}_1& q_2& \dots & q_n\end{array}\right]\end{array}$$

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It then follows that

$$G(s)=\bar{C}(s\mathbb{I}-\Lambda {)}^{-1}\hat{B}+\bar{D}$$

Where

$$(s\mathbb{I}-\Lambda {)}^{-1}=\left[\begin{array}{cccc}\frac{1}{s-{\lambda }_1}& 0& \dots & 0\\ 0& \frac{1}{s-{\lambda }_2}& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \frac{1}{s-{\lambda }_n}\end{array}\right]$$