After finishing some exercises, it seems that an $n\times n$ matrix $A$ will also have a $n\times n$ jordan form. Is my observation correct? If this is a common phenomenon, could you explain reasons behind it?
I guess it is about the dimension of eigenspaces, but I am not sure.
The Jordan form of a matrix is just the matrix transformed by a change of basis. They both represent the same linear map from an $n$-dimensional vector space to another, so they are of course both $n\times n$.