Classifying Complex Matrices with Cubed Identity: What Does Similarity Mean?

[CoachZ]

The question posed is "Classify up to similarity all 3 x 3 complex matrices A s.t. A^{3} = I. I think the biggest problem I'm having is understanding what exactly this is asking me to do. The part that says "Classify up to similarity" is really throwing me off, so if someone could tell me what that implies, it would be very helpful!

Maybe it deals with equivalence classes, or something like that?

 

[tiny-tim]

… The part that says "Classify up to similarity" is really throwing me off, so if someone could tell me what that implies, it would be very helpful!

Maybe it deals with equivalence classes, or something like that?

Hi CoachZ!

Yes, it's putting two matrices in the same equivalence class if they are similar.

 

[hamster143]

It means, can you find a set of complex 3x3 matrices s.t. A^3 = I, such that none of them are similar to each other, and any other such 3x3 matrix is similar to one of the matrices in your set?

Hint: what can you say about eigenvalues of A?

 

[CoachZ]

It means, can you find a set of complex 3x3 matrices s.t. A^3 = I, such that none of them are similar to each other, and any other such 3x3 matrix is similar to one of the matrices in your set?

Hint: what can you say about eigenvalues of A?

I think that I'm a little confused, because if A^3 = I, then the eigenvalues for A would simply be the eigenvalues for I, which is just 1,1,1, since I is a diagonal matrix. Therefore, the characteristic polynomial of such A must be in the form (x-1)^3, right? Is there a way to compose matrices from characteristic polys?

 

[g_edgar]

I think that I'm a little confused, because if A^3 = I, then the eigenvalues for A would simply be the eigenvalues for I,

No, the eigenvalues must have cube equal to 1, but there are complex numbers other than 1 itself with that property...

Therefore, the characteristic polynomial of such A must be in the form (x-1)^3, right?

No, for example the characteristic polynomial could be x^3-1 .

Is there a way to compose matrices from characteristic polys?

One would assume the textbook would have such information before asking this question.