What is the necessary condition for matrix commutation?

In summary, the necessary condition for two arbitrary matrices, A and B, to commute is that they must both be square and simultaneously triangularisable. Additionally, if the scalar field is not algebraically closed, the matrices must also be simultaneously triangularisable under the complex numbers. This is essentially the only way for two matrices to commute.
  • #1
fairy._.queen
47
0
Hi all!

I was wondering what the necessary condition is for two arbitrary matrices, say A and B, to commute: AB = BA.

I know of several sufficient conditions (e.g. that A, B be diagonal, that they are symmetric and their product is symmetric etc), but I can't think of a necessary one.

Thanks in advance!
 
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  • #3
Ok, so it seems the condition (quite sensible actually!) is that they must both be square and simultaneously triangularisable. Thanks a lot!
 
  • #4
What happens, though, when the matrix scalar field is not algebraically closed? I'm happy with the fact that, in this case, if the two matrices are diagonalisable and commute then they are simultaneously diagonalisable, but what is a necessary condition for arbitrary, say, real matrices to commute (even when they can't be diagonalised)?

Thanks!
 
  • #5
you could just pretend that the scalar field is the complex numbers, and see if you can make the matrices simultaneously triangular under the complex numbers. If they are not simultaneously triangular under the complex numbers, they will not commute. And if they are simultaneously triangularizable under the complex numbers, then they do commute.
 
  • #6
Ok, it makes sense. Thanks a lot!
 
  • #7
fairy._.queen said:
I was wondering what the necessary condition is for two arbitrary matrices, say A and B, to commute: AB = BA.

This remark is from a bound set of notes I found in a used book store: "If two matrices are both polynomials in the same matrix [itex]C [/itex], then they commute. As we shall see later, this is essentially the only way in which we can have commuting matrices."
 

1. What is matrix commutation?

Matrix commutation refers to the ability of two matrices to be multiplied in either order and produce the same result. In other words, the order of multiplication does not change the outcome.

2. Why is matrix commutation important?

Matrix commutation is important in various fields of mathematics and science, such as linear algebra, quantum mechanics, and electrical engineering. It allows for simpler calculations and more efficient problem solving.

3. What is the necessary condition for matrix commutation?

The necessary condition for matrix commutation is that the two matrices must be square and of the same dimension. This means that they must have the same number of rows and columns.

4. What happens if the necessary condition for matrix commutation is not met?

If the two matrices do not have the same dimension, they will not be able to be multiplied in either order and produce the same result. This is known as non-commutativity and can lead to more complex calculations and solutions.

5. Can non-square matrices commute?

No, non-square matrices cannot commute. The necessary condition for matrix commutation is that the matrices must be square, meaning they have the same number of rows and columns. Non-square matrices have different dimensions and therefore cannot commute.

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