Meaning of minimal polynomial | Babel Free

Definitions

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.
2. 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.

Examples

“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 Tⁱ, without actually computing the powers of T is indicated in the solution to Problem 5.9.”

“In this section we will show that if #92;#123;A,B,C,D#92;#125; is a controllable and observable realization of #92;mathbfG, then #92;lambda is a pole of #92;mathbfG if and only if #92;lambda is an eigenvalue of A. Moreover, the roots (multiplicities included) of the minimal polynomial of A are the poles of #92;mathbfG.”

“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.”

“[…]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.”

“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.”

“Let #92;alpha be an algebraic number with degree n and let P be the minimal polynomial for #92;alpha (see Section 6.5).”

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