9. Simultaneous reduction
By this we mean any result that reduces two or more endomorphisms with the same transition matrix.
9.1 Case of two matrices
The best-known is that if A and B are two commutating complex matrices, then they have a common eigenvector and, consequently, there is a common basis in which they both have the triangular form. Any matrix of the form P (A,B ) (AB – BA),
where P is a (non-commutative) polynomial in two variables, is then nilpotent.
The following theorem sheds more light on the subject.
Theorem 7. If P (A,B )(AB – BA) is nilpotent for any polynomial P, then A and B are simultaneously trigonalizable (see reference in
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Simultaneous reduction
Another view of Jordanization
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