Is there any way to tell if a matrix is similar to a matrix other than finding an explicit such that ?
The reason is that I am trying to find conjugacy classes of in the easiest possible way.
Similar matrix - Wikipedia, the free encyclopedia
Jordan normal form - Wikipedia, the free encyclopedia
Matrices A and B are similar iff their Jordan normal forms are equal (this is because every matrix in an algebraically closed field has a unique Jordan form).
If you know the minimal polynomial of a matrix, then it's easier to calculate its jordan form:
Let be an algebraically-closed field, a vector space over ,
represent a linear operator
be the minimal polynomial of A, where each is the geometrical multiplicity of , and let
Then by the primary decomposition theorem, and after a few more steps, we get that each is represented by a Jordan block matrix:
And the Jordan canonical form of A is the direct sum of its Jordan blocks.
In general, other poster already told you: two square matrices over some field are simmilar iff their Jordan forms (over some extension field of the original one that contains all the eigenvalues of either of the matrices) are equal.
In you case, though, it's way easier since you're dealing with 2 x 2 matrices, and then two matrices here are simmilar iff they have the same characteristic and minimal polynomials, which is far from being true in other cases (well, also with 3x3 matrices it is true, but with 4 x 4 there already counterexamples)
And both matrices above have the same char. pol. and the same min. pol.
What's important to remember here? The power at which every single irreducible factor of the min. pol. is raised only tells us what's the maximum size of a Jordan block corresponding to that eigenvalue and that there's at least one Jordan block wrt this eigenvalue of this size, but it does NOT necessarily tells us how many blocks of this size are there (for that we already need the dimension of the corresponding eigenspace)...unless the matrix is of order at most 3 x 3.