Constructing Eigenvectors from Commuting Matrices: A Unique Classification

[greisen]

Hey all,

I have two matrices A,B which commute than I have to show that these eigenvectors provide a unique classification of the eigenvectors of H?

From these pairs of eigenvalue is it possible to obtain the eigenvectors?


I don't quite know how to procede any suggestions?

Thanks in advance

 

[StatusX]

What is H?

 

[HallsofIvy]

Hey all,

I have two matrices A,B which commute than I have to show that these eigenvectors provide a unique classification of the eigenvectors of H?

This makes no sense. Obviously, the eigenvectors of A and B will tell you nothing about the eigenvectors of some arbitrary third matrix H. What relationships are there between A, B, and H?

From these pairs of eigenvalue is it possible to obtain the eigenvectors?


I don't quite know how to procede any suggestions?

Thanks in advance

Yes. Look closely at how A and B are related to H.

 

[greisen]

Sorry not H but A the same matrix

 

[StatusX]

So you're asking if the eigenvectors of B determine the eigenvectors of A, given that A and B commute? This doesn't sound right, since the identity matrix commutes with everything. You can narrow down the possible eigenvectors of A, but you won't get a "unique classification."

 

[greisen]

to see if I understand correctly - let's assume that the matrix A har the eigenvalues {1,2,2} and the matrix B has the eigenvalues {-1,1,1} - then it is possible to construct the eigenvectors of B according to the common unique pairs of A and B( (1,1),(2,1),(2,-1)) giving the following eigenvectors: (1,0,0) , (0,1,1) , (0,-1,1) ?

And had it not been possible with unique pairs of eigenvalues would the eigenvectors not be orthogonanle?

Thanks in advance