The minimal and characteristic polynomials of a linear transformation encode much useful information about the transformation. Of course, they do not encode all useful information -- for example, we know that two linear transformations are conjugate if and only if they have the same minimal (or characteristic) polynomial and the powers of the prime factors of the polynomial applied to the transformations have the same ranks. On the other hand, the matrix of the transformation with respect to a basis (in particular a normal form) encodes all information.
Are there any other "natural" invariant objects associated with a linear transformation, which encode more information than the minimal and characteristic polynomials and are straightforward to study, other than matrix forms?