Proving Non-Conjugate Matrices with Characteristic and Minimal Polynomials

[Steph]

I have a question where I must prove that none of the following matrices are conjugate:

1 1 0
0 1 1
0 0 1

1 1 0
0 1 0
0 0 1

1 0 0
0 1 0
0 0 1


I started by working out the characteristic polynomials since if these were different, the matrices couldn't be conjugate. But all three have the same characteristic polynomial [(1-t)^3], so that didn't help me.

I'm not sure what else I can do from here. I know that if A and B are conjugate, A = QBP where Q is the inverse of the matrix P. Also, matrices are conjugate if they represent the same linear transformation after a change of basis.

But I'm not sure how to apply these definitions to a specific example.

Thanks in advance for any help.

 

[matt grime]

minimal polynomial, not characteristic polynomial.another way (equivalent) is to think of some other invariants under conjugation such as the number of eigenvectors.

 

[Steph]

I have written in my notes that conjugate matrices have the same characteristic polynomial -- am I correct if I say that conjugate matrices must have the same characteristic polynomial, but that matrices with the same characteristic polynomial are not necessarily conjugate? Is the point here that the minimal polynomials must also be the same for conjugate matrices, and that this is a more useful condition?


Many thanks for your help - I've used your suggestion and completed that and the rest of the question.

 

[matt grime]

Of course conjugate matrices have the same char poly, but as you noted yourself that is not enough to differentiate between non-conjugate matrices. Minimal polys do differentiate between them.