8. Calculating the switch dimension

This calculation is used to compute the dimension of orbits. The method consists in finding the dimension of its commutant for a nilpotent matrix, as this is sufficient given the following lemma.

Lemma 2. If $M = [\begin{matrix} A \\ 0 \end{matrix} \begin{matrix} 0 \\ B \end{matrix}]$ is a block-diagonal matrix verifying that the characteristic polynomials of A and B are prime to each other, then the matrices that commute with M are block-diagonal, i.e. $[\begin{matrix} X \\ 0 \end{matrix} \begin{matrix} 0 \\ Y \end{matrix}]$ ...

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Calculating the switch dimension

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page Special families of matrices. Matrices of class δ

Simultaneous reduction

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References

(1) - ADKINS (A.), WEINTRAUB (S.) - Algebra. An Introduction via module Theory, - Springer 1992. ISBN 0-387-97839-9.
(2) - CHAMBADAL (L.), OVAERT (J.-L.) - Algèbre linéaire et algèbre tensorielle - , Dunod 1968.
(3)...

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