Elementary questions on minimal polynomials of linear transformations

Question

1. Is the minimal polynomial of a linear transformation equal to that of its matrix representation on some basis?
2. One of my textbooks on linear algebra defines eigenvalues and eigenvectors in terms of minimal polynomials, while the other textbooks define them in terms of characteristic polynomials. What is good about defining them in terms of minimal polynomials?
3. Why are minimal polynomials important?

Answer 1

1. The minimal polynomial of a linear transformation is equal to the minimal polynomial of its matrix representation with respect to any basis. That is, if $\beta$ is any basis, and $M=[T]_{\beta}$ is the matrix representation of $T$ relative to $\beta$, then the minimal polynomial of $T$ is equal to the minimal polynomial of $M$.

2. There is nothing good or bad either way. It is a theorem that every irreducible factor of the characteristic polynomial divides the minimal polynomial, so both definitions amount to the same thing in the end. The difference is just which of the two concepts you consider to be "earlier". In 'traditional' linear algebra books (that are based on matrices and determinants), the characteristic polynomial is simpler/earlier than the minimal polynomial, so it makes sense to define the eigenvalues in terms of the characteristic polynomial. (In fact, most often the characteristic polynomial is deduced from properties that eigenvalues have to have: you note that $\lambda$ is an eigenvalue if and only if $T-\lambda I$ is not one-to-one, if and only if $T-\lambda I$ is not invertible, if and only if $\det(T-\lambda I)=0$, and from there you define the characteristic polynomial as an interesting objects). On the other hand, from the point of view of module theory, the minimal polynomial is more natural and so if you are taking the point of view of thinking of linear transformations as defining module structures on $V$, then the minimal polynomial is a more reasonable starting points.

3. Minimal polynomials carry more information than characteristic polynomials; not only do they give you the eigenvalues (like the characteristic polynomials do), they also give you the size of the largest Jordan block associated to any given eigenvalue in the Jordan canonical form of the linear transformation, the size of the largest companion block associated to any given irreducible factor in the rational canonical form, a necessary and sufficient condition for diagonalizability ($T$ is diagonalizable if and only if the minimal polynomial of $T$ splits and has no repeated roots), and more besides.