Similarity and commuting matrices

Question

Is there any connection between similar matrices and matrices that commute with each other? That is, for commuting matrices $AB = BA$ is there some sort of similarity relationship between $A$ and $B$, i.e. is $A = PBP^{-1}$ for some matrices $P,P^{-1}$?

Answer 1

No not necessarily. Let $A$ be the identity matrix and $B$ be the zero matrix. They commute but there is no invertible matrix such that $A = PBP^{-1}$.

Answer 2

No, for example $I$ and $2I$ commute but are certainly not similar.

Answer 3

I think the matrices having similarity is a very special type of commuting matrix. Because if two matrices have similarity, that means they should be able to be diagonalized, so the have to be a squarematrix and have full rank, because I don't want the diagonal matrix produced having zero entries on the diagonal. When we admit this is true, then we can see A as a matrix expression of linear transformation from a basis to the smae basis in a given space, so do B, and the P matrix and its inverse that you multiply on both sides can be seen as two corresponding matrix make the linear trnsformation goes from another basis to itself. When A and B are similar matrix, you will find that A and B are actually doing the similar things to a set of basis which is known as their eigenvector. So A and B are just making their eigenvector longer or shorter, so their operation sequence can change, and the things they do on the basis on the eigenvector is the same for A and p-1Bp, so they are similar. But what about commuting matrices, if I suppose A and B are all full rank and can be diagonalized, it means their behaviours can be shown by the operation on their eigenvvectors, so when does A and B can change their operating turn? I think that's when A and B can find a same set of eigenvector, so they are just changing the length not the orientation of the eigenvactors. So I think when A and B has the same eigenvector, they are commuting matrices, however, when we find the things the do to the same eigenvector is the same then A and B is a set of similar matrix, So I will say similar matrix is a special type of commuting matrix.