minimal polynomial

minimal polynomial (plural minimal polynomials)

1. (linear algebra) 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 divides the characteristic polynomial. Also, any root of the characteristic polynomial is also a root of the minimal polynomial, so the two kinds of polynomial have the same roots, only (possibly) differing in their multiplicities.)

   • 1965 [John Wiley], Robert B. Ash, Information Theory, 1990, Dover, page 161,

     A procedure for obtaining the minimal polynomial of the matrix ${\displaystyle T^{i}}$ , without actually computing the powers of ${\displaystyle T}$  is indicated in the solution to Problem 5.9.

   • 2003, Martin J. Corless, Art Frazho, Linear Systems and Control: An Operator Perspective, Marcel Dekker, page 77:

     In this section we will show that if ${\displaystyle \{A,B,C,D\}}$  is a controllable and observable realization of ${\displaystyle \mathbf {G} }$ , then ${\displaystyle \lambda }$  is a pole of ${\displaystyle \mathbf {G} }$  if and only if ${\displaystyle \lambda }$  is an eigenvalue of ${\displaystyle A}$ . Moreover, the roots (multiplicities included) of the minimal polynomial of ${\displaystyle A}$  are the poles of ${\displaystyle \mathbf {G} }$ .

   • 2007, A. R. Vasishta, Vipin Vasishta, A.K. Vasishta, Abstract and Linear Algebra, Krishna Prakashan Media, 3rd Edition, page CA-439,

     Theorem 1. The minimal polynomial of a matrix or of a linear operator is unique.

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.
   • 2005, Victor Shoup, A Computational Introduction to Number Theory and Algebra, Cambridge University Press, page 438:

     […] we are given an element ${\displaystyle \alpha \in E}$ , and want to compute the minimal polynomial ${\displaystyle \phi \in F[X]}$  of ${\displaystyle \alpha }$  over ${\displaystyle F}$ .

   • 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:

     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 equal to the order of this element provided the highest weight of this element is large enough with respect to the ground field characteristic.

   • 2012, Alan Baker, A Comprehensive Course in Number Theory, Cambridge University Press, page 61:

     Let ${\displaystyle \alpha }$  be an algebraic number with degree ${\displaystyle n}$  and let ${\displaystyle P}$  be the minimal polynomial for ${\displaystyle \alpha }$  (see Section 6.5).

• annihilator

•   Minimal polynomial (field theory) on Wikipedia.Wikipedia
•   Minimal polynomial (linear algebra) on Wikipedia.Wikipedia
•   Characteristic polynomial on Wikipedia.Wikipedia
  •   Eigenvalue on Wikipedia.Wikipedia
• Matrix Minimal Polynomial on Wolfram MathWorld
• Algebraic Number Minimal Polynomial on Wolfram MathWorld
• minimal polynomial PlanetMath.org