Commuting matrices have common eigenvalues

[Grand]

Homework Statement

How do we prove that commuting matrices have common eigenvalues?

Homework Equations

The Attempt at a Solution

 

[Mark44]

Homework Statement

How do we prove that commuting matrices have common eigenvalues?

Start by writing some mathematics, using the definition of commuting matrices and what it means for \lambda to be an eigenvalue of a matrix.

 

[Grand]

OK, I've benn trying this for some time now, but:

common eigenvalues means that:
det(A-\lambda I)=det(B-\lambda I)=0

and we have to prove that AB=BA

 

[Mark44]

You have it backwards. You are given that A and B commute, and need to show that any eigenvalue of A is also an eigenvalue of B.

 

[Grand]

OK, I agree. But where do I start. I tried to add expressions to both sides of AB=BA:

AB=BA/\lambda
\lambda AB=\lambda BA/-B
(\lambda A-A)B=\lambda BA-B

but I'm not really going anywhere

 

[Mark44]

OK, I agree. But where do I start. I tried to add expressions to both sides of AB=BA:

AB=BA/\lambda
\lambda AB=\lambda BA/-B

What do the equations above mean?

(\lambda A-A)B=\lambda BA-B

but I'm not really going anywhere

If \lambda is an eigenvalue for a matrix A, then for some nonzero vector x,
Ax = \lambdax.

 

[Grand]

So how do we use this?