Determining set of similarity classes for matrix?

[cookiesyum]

Homework Statement

Determine the set of similarity classes of 3x3 matrices A, over C, which satisfy A^3 = 1. Determine..etc etc...which satisfy A^6 = 1.

Determine the set of similarity classes of 6x6 matrices A, over C, with characteristic polynomial:

char(A,x) = (x^4 -1)(x^2 - 1)

Homework Equations

The Attempt at a Solution

To find similar matrices, you probably have to find the rational, or jordan, or diagonal matrices...? I have no idea how to start these problems. =/

 

[mighty2000]

Thank you for your post. To find the set of similarity classes for 3x3 matrices A with A^3 = 1, we can start by considering the eigenvalues of A. Since A^3 = 1, the eigenvalues must satisfy the equation λ^3 = 1. This means that the possible eigenvalues are 1, ω, and ω^2, where ω is a complex cube root of unity (ω = e^(2πi/3) or ω = e^(4πi/3)).

Now, we can use the fact that similar matrices have the same eigenvalues to determine the set of similarity classes. For example, if A has eigenvalue 1, then it is similar to the identity matrix. If A has eigenvalue ω, then it is similar to a matrix with 1 in the first diagonal entry, ω in the second diagonal entry, and ω^2 in the third diagonal entry. Similarly, if A has eigenvalue ω^2, then it is similar to a matrix with 1 in the first diagonal entry, ω^2 in the second diagonal entry, and ω in the third diagonal entry.

Therefore, the set of similarity classes for 3x3 matrices with A^3 = 1 is the set of all diagonal matrices with entries 1, ω, and ω^2 in any order.

For the second part of the question, we can use a similar approach. Since the characteristic polynomial has factors (x^4 - 1) and (x^2 - 1), the eigenvalues must satisfy the equations λ^4 = 1 and λ^2 = 1. This means that the possible eigenvalues are 1, -1, i, and -i. Using the same logic as before, we can determine that the set of similarity classes for 6x6 matrices with the given characteristic polynomial is the set of all diagonal matrices with entries 1, -1, i, and -i in any order.

I hope this helps you with your homework. Good luck!