Find all $A,B$ such that $AB-BA=0$.

Question

Can someone give me a hint how to find all $n\times n$ matrices $A,B$ over an arbitrary field, such that they commute, i.e. such that $AB=BA$ ?

I found this problem in some lectures notes where the trace of a matrix was discussed, so my guess is, that I should use it - but I have no idea how (although I know that $tr(AB)=tr(BA)$). Manually writing this out and trying to solve a huge system just doesn't seem like a good strategy.

$tr(AB)=tr(BA)$ this is always true for any $A$ and $B$ even if $AB\ne BA$ so I don't think this will be much helpful here. — i.a.m, Mar 06 '13 at 22:21
see my answer at http://math.stackexchange.com/questions/92480/given-a-matrix-is-there-always-another-matrix-which-commutes-with-it That is, it is a bit optimistic to try to describe all pairs. It gets almost manageable if you fix $A$ and then look for those $B$ that commute with it. — Will Jagy, Mar 06 '13 at 22:24

score 3 · Answer 1 · answered Mar 06 '13 at 22:24

3

You can use the following necessary condition to aid your search:

Take the algebraic closure $ \overline{\mathbb{F}} $ of the given field $ \mathbb{F} $. If $ A $ and $ B $ commute, then $ A $ and $ B $ are simultaneously triangularizable in $ {\mathbf{M}_{n}}(\overline{\mathbb{F}}) $.

answered Mar 06 '13 at 22:24

Haskell Curry

20,144

4

Note: The converse is false, however. – Haskell Curry Mar 06 '13 at 22:31
3

While this condition is certainly necessary, I don't immediately see how it helps to answer the question, i.e., to actually find all commuting pairs of matrices. – Marc van Leeuwen Mar 07 '13 at 06:25

score 1 · Answer 2 · answered Mar 06 '13 at 22:53

1

You can transform $A$ into the Jordan normal form, write $B$ in block form similar to $A$, multiply blocks etc. This method of solution of matrix equations is described detailly in

F.Gantmacher. Theory of matrices, AMS Chelsea publishing, 1959.

answered Mar 06 '13 at 22:53

Boris Novikov

17,754

Find all $A,B$ such that $AB-BA=0$.

2 Answers2

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