Similarity classes for matrices

[hypermonkey2]

Homework Statement

How would we find number of similarity classes for a nxn matrix over the field Fp (cyclic of order p) for n=2,3,4?

Homework Equations

A and B are similar iff they have the same monic invariant factors

The Attempt at a Solution

I would say 4 classes for n=2, since all that is really important is if the characteristic polynomial factors over Fp or not. Should the answer depend on p? Or is this just taking cases?
Thanks for any help!

 

[mighty2000]

Hello,

The number of similarity classes for a nxn matrix over the field Fp can be determined by looking at the number of distinct monic invariant factors of the characteristic polynomial of the matrix. This is because two matrices A and B are similar if and only if they have the same monic invariant factors.

For n=2, we can have a maximum of 2 distinct monic invariant factors, namely the characteristic polynomial and the constant polynomial. Therefore, there can be a maximum of 2 similarity classes for a 2x2 matrix over the field Fp.

For n=3, we can have a maximum of 3 distinct monic invariant factors, namely the characteristic polynomial, a linear polynomial and the constant polynomial. Therefore, there can be a maximum of 3 similarity classes for a 3x3 matrix over the field Fp.

For n=4, we can have a maximum of 4 distinct monic invariant factors, namely the characteristic polynomial, a quadratic polynomial, a linear polynomial and the constant polynomial. Therefore, there can be a maximum of 4 similarity classes for a 4x4 matrix over the field Fp.

The answer does not depend on p, as the number of similarity classes is determined by the number of distinct monic invariant factors, which is a property of the matrix itself and not the field. I hope this helps!