Explain Minimal Polynomial and Generating Polynomial.
Answers
In linear algebra the minimal polynomial of an algebraic object is the monic polynomial of least degree which that object satisfies. Examples include the minimal polynomial of a square matrix, an endomorphism of a vector space or an algebraic number.
The general setting is an algebra A over a field F. We give A the structure of a module over the polynomial ring F[X] by defining the action of f(x) = \sum_{n=0}^d f_i X^i on a to be f(a) = \sum_{n=0}^d f_i a^i where a0 is defined to be the unit element of A.
We say that f "annihilates" a, or that a "satisfies" f, if f(a) = 0. The set of polynomials that annihilate a given element a forms an ideal ann(a) in F[X], which is a Euclidean domain. Hence the annihilator ideal is a principal ideal with the minimal polynomial as monic generator.
The minimal polynomial may also be defined as the polynomial of least degree which annihilates a: it then has the property that it divides any other polynomial which annihilates a.