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I'm working in the text "Introduction to Applied Nonlinear Dynamical Systems and Chaos"and I have a doubt in the following exercise

Consider a third-order autonomous vector field near a fixed point having linear part $$\left(\begin{matrix} 0&-\omega&0\\ \omega&0&0\\ 0&0&0 \end{matrix}\right)$$ find the normal form. (Hint: Lump the two coordinates associated with the block$$\left(\begin{matrix} 0&-\omega\\ \omega&0 \end{matrix}\right)$$ into the single complex coordinate.)

My doubt is what means: Lump the two coordinates associated with the...

Thanks in advance.

Skinner.
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  • Hint: Complex numbers can be represented as real matrices that obey all of the same operations by the following (field) isomorphism: $$a+bi \leftrightarrow \begin{pmatrix}a & -b \ b & a\end{pmatrix}$$ – Ninad Munshi Nov 26 '22 at 14:21

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