Modal Form

Opposed to Jordan Form, this allows us to deal with only real number if we have pairs of complex conjugated eigenvalues.

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The Transformation Matrix $M$ is given in the following Similarity Transform

$${\Lambda }_m=M^{-1}\Lambda M$$

Where

$$\begin{array}{rl}\Lambda & =\left[\begin{array}{cccccc}{\lambda }_1& 0& 0& 0& 0& 0\\ 0& & {\alpha }_1+i{\beta }_1& 0& 0& 0\\ 0& 0& 0& {\alpha }_1-i{\beta }_1& 0& 0\\ 0& 0& 0& 0& {\alpha }_2+i{\beta }_2& 0\\ 0& 0& 0& 0& 0& {\alpha }_2-i{\beta }_2\end{array}\right]\end{array}$$

And

$$M=\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& \frac{1}{2}& -\frac{i}{2}& 0& 0& 0\\ 0& \frac{1}{2}& \frac{i}{2}& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& \frac{1}{2}& -\frac{i}{2}\\ 0& 0& 0& 0& \frac{1}{2}& \frac{i}{2}\end{array}\right]$$ $$M^{-1}=\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 1& 1& 0& 0& 0\\ 0& i& -i& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 1\\ 0& 0& 0& 0& i& -i\end{array}\right]$$