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 = $\lambda$x.

 

[Grand]

So how do we use this?