minimal polynomial

短语

[计] 最小多项式

词形变化

minimal polynomials 复数 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.

数学

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.

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.

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.