Minimal polynomials and characteristic polynomials?

Question

I am trying to understand the similarities and differences between the minimal polynomial and characteristic polynomial of Matrices. 1. When are the minimal polynomial and characteristic polynomial the same 2. When are they different 3. What conditions (eigenvalues/eigenvectors/...) would imply 1 or 2 4. Please tell me anything else about these two polynomials that is essential in comparing them.

Answer 1

This will not be a complete answer, and probably any complete answer should include examples.

The fact that a matrix is a root of its own characteristic polynomial (the Cayley–Hamilton theorem) means there is at least one polynomial of which the matrix is a root.

Degrees of polynomials are in the set { 0, 1, 2, 3, . . . }, and that is a well-ordered set, meaning every non-empty subset of it has a smallest member. The set of degrees of polynomials having the matrix as a root is non-empty because the degree of the characteristic polynomial belongs to it, by the Cayley–Hamilton theorem.

Therefore there is a smallest such degree. Call that number m. There is some polynomial p(λ) of degree m that has our matrix M as a root, i.e. we have p(M) = 0. Dividing both sides of that equation by its leading coefficient yields a monic polynomial, i.e. a polynomial with leading coefficient 1, of degree m, with M as a root. Now one would like to prove there cannot be more than one such monic polynomial. Suppose p(λ) and q(λ) are two monic polynomials of degree m with M as a root. Then p(λ) – q(λ) has M as a root. But the leading terms of p(λ) and q(λ) are the same, both being λ^m, so they cancel out when one subtracts. This gives us a polynomial whose degree is less than m. But no polynomial of degree less than m can have M as a root because, as seen above, m is the smallest. Therefore there can be only one minimal polynomial.

Answer 2

So if the characteristic polynomial factors i.e. (a_1-lambda)^(n_1)..(a_2-lambda)^(n_2)... then the minimal is just any combination of that factor with least power (a_1-lambda)^(m_1)..(a_2-lambda)^(m_2)... such that 1<= m <= n.

There are many different matrices, and each have unique cases to inspect by using such methods as polynomials. The polynomial itself can tell you 'characteristics' of the matrix e.g. if it is diagonalizable, ...

Best thing to do is research how the different polynomials tell you characteristics about the matrix e.g. a singular matrix, symmetric, triangular, complex, ...

Remember that as a STEM student, you are being trained to do research so best thing to do is learn how to research this stuff on your own because that is what you will be doing in a job setting.

Hope that helps a little.

JJ