How to make a matrix B such that AB=BA

[leon1127]

B = 0
B = I
B = A
B = A^-1

There are more I suppose

 

[DeadWolfe]

enough already! this is a boring question!

here is a more interesting one: prove that a holomorphic map of a riemann surface to itself that induces the identity on homology is the identity map.

Is our surface compact?

 

[navidnfk]

I have a Idea How to Find B that AB=BA!

 

[HallsofIvy]

Without knowing A?

 

[trambolin]

hotcommodity already gave the answer, here is another possibility... take a unitary matrix, and let B = A^T plug it in, done !

but if you are desperately looking for numerical values, you can try the following MATLAB code for almost commuting (!?) matrices,

A = rand(5);
B = lyap(A,-A,1e-15*eye(5));
A*B-B*A

I have to perturb the Sylvester equation(since because otherwise it will give the trivial answer B=0. Or you can come up with a automated code that forms the \hat{A}b=0 equation where b is the vectorized entries of B. It is not hard it is just tedious. It will be as

\left\{(I\otimes A) + (-A^T\otimes I)\right\} b= 0

etc... But numerically you should still perturb that and this is not the best way in terms of numerical stability. My favorite will be semidefinite programming... Dig in if you like...

Conclusion, a lot of choices, so pick one, or find the conditions and teach me those.

 

[hotcommodity]

hotcommodity already gave the answer, here is another possibility...

I didn't give the answer (not the first one anyways).

I don't want to get in trouble again, haha

 

[JasonRox]

I stand by matt_grime. It's frustrating although I'm probably that person from time to time.

matt_wonk, that's a boring question. Um...