minimal polynomial

Noun

minimal polynomial (plural minimal polynomials)

1. For a given square matrix M over some field K, the smallest-degree monic polynomial over K which, when applied to M, yields the zero matrix.
   • Each root of the minimal polynomial of a matrix M is an eigenvalue of M and a root of its characteristic polynomial. (A root of the minimal polynomial has a multiplicity that is less than or equal to the multiplicity of the same root in the characteristic polynomial. Thus the minimal polynomial...
   • 1965 [John Wiley], Robert B. Ash, Information Theory, 1990, Dover, page 161,
     A procedure for obtaining the minimal polynomial of the matrix Tⁱ, without actually computing the powers of T is indicated in the solution to Problem 5.9.
2. (field theory) Given an algebraic element α of a given extension field of some field K, the monic polynomial of smallest degree of which α is a root.
   • […]we are given an element #92;alpha#92;inE, and want to compute the minimal polynomial #92;phi#92;inF#91;X#93; of #92;alpha over F.2005, Victor Shoup, A Computational Introduction to Number Theory and Algebra, Cambridge University Press, page 438
   • The minimal polynomials of the images of unipotent elements in irreducible rational representations of the classical algebraic groups over fields of odd characteristic are found.[…]It occurs that for each unipotent element the degree of its minimal polynomial in an irreducible representation is...2009, Irina D. Suprunenko, The Minimal Polynomials of Unipotent Elements in Irreducible Representations of the Classical Groups in Odd Characteristic, American Mathematical Society, page 1