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
Exclusive to subscribers. 97% yet to be discovered!
Already subscribed? Log in!
Simultaneous reduction
Another view of Jordanization
Article included in this offer
"Mathematics"
(
166 articles
)
Updated and enriched with articles validated by our scientific committees
A set of exclusive tools to complement the resources
References
Exclusive to subscribers. 97% yet to be discovered!
Already subscribed? Log in!
