A polynomial P is annihilating or called an annihilating polynomial in linear algebra and operator theory if the polynomial considered as a function of...

minimal polynomial μA of an n × n matrix A over a field F is the monic polynomial P over F of least degree such that P(A) = 0. Any other polynomial Q with...

p_{A}(A)=\mathbf {0} ;} that is, the characteristic polynomial p A {\displaystyle p_{A}} is an annihilating polynomial for A . {\displaystyle A.} One use for the...

A polynomial annihilates A {\displaystyle A} if p ( A ) = 0 {\displaystyle p(A)=0} ; p {\displaystyle p} is also known as an annihilating polynomial. Thus...

obtained from the characteristic polynomial of ⁠ A {\displaystyle A} ⁠ or, more generally, from any annihilating polynomial of ⁠ A {\displaystyle A} ⁠. This...

dividing one such polynomial by another, when each has a finite order (highest exponent), results in an infinite-order polynomial. An annihilator operator, denoted...

{\displaystyle f'(x)} . A {\displaystyle A} is called an annihilating operator of f (the annihilating operators of f {\displaystyle f} form an ideal in the...

consider the family of polynomials which annihilates an operator T {\displaystyle T} . This family is an ideal in the ring of polynomials. Furthermore, it is...

{\displaystyle f_{k}(X)} is the monic polynomial of smallest degree annihilating the module, or zero if no such non-zero polynomial exists. In the first case dim...

1007/s00209-003-0490-6. Lewis, D. W.; McGarraghy, S. (2000). "Annihilating polynomials, étale algebras, trace forms and the Galois number". Archiv der...

extending the field F (whence the "rational"), notably without factoring polynomials, this shows that whether two matrices are similar does not change upon...

Third, Q {\displaystyle Q} should take a polynomial in x {\displaystyle x} and p {\displaystyle p} to a "polynomial" in X {\displaystyle X} and P {\displaystyle...

Alternatively, the set of polynomials that annihilate a given A form an ideal I in C[x], the principal ideal domain of polynomials with complex coefficients. The...

commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers Z...

commutative algebra. Prominent examples of commutative rings include polynomial rings, rings of algebraic integers, including the ordinary integers Z...

characteristic polynomial of A. So the algebraic multiplicity is the multiplicity of the eigenvalue as a zero of the characteristic polynomial. Since any...

is an element of the field of fractions of A that is a root of a monic polynomial with coefficients in A, then x is itself an element of A. Many well-studied...

given binary output sequence. The algorithm will also find the minimal polynomial of a linearly recurrent sequence in an arbitrary field. The field requirement...

commutatively (which is annihilated by the derived algebra). Thus of central interest are the free commutative algebras, namely the polynomial algebras. In this...

expanding on the work of Sato and Joseph Bernstein on the Bernstein–Sato polynomial. Early major results were the Kashiwara constructibility theorem and Kashiwara...

may be computed in terms of division polynomials. Analytic torsion Arithmetic dynamics Flat module Annihilator (ring theory) Localization of a module...

For example, take an irreducible polynomial f ( x 1 , … , x n ) {\displaystyle f(x_{1},\ldots ,x_{n})} in a polynomial ring F [ x 1 , … , x n ] {\displaystyle...

solution, one may write the unknown polynomials as polynomials with unknown coefficients. Then, as two polynomials are equal if and only if their coefficients...

theorem was first proven by Emanuel Lasker (1905) for the special case of polynomial rings and convergent power series rings, and was proven in its full generality...

are met: c i {\displaystyle c_{i}} are constants. g(x) is a constant, a polynomial function, exponential function e α x {\displaystyle e^{\alpha x}} , sine...

said to be integral over a subring A of B if b is a root of some monic polynomial over A. If A, B are fields, then the notions of "integral over" and of...

theory itself and for its applications, such as homological properties and polynomial identities. Commutative rings are much better understood than noncommutative...

{Supp} M\cong \operatorname {Spec} (R/I)} , the vanishing locus of the polynomial f. Looking at the short exact sequence 0 → I → R → R / I → 0 {\displaystyle...

any. The solutions of homogeneous linear differential equations with polynomial coefficients are called holonomic functions. This class of functions is...

{\displaystyle A^{n}} is diagonalizable, then A {\displaystyle A} is annihilated by some polynomial ( x n − λ 1 ) ⋯ ( x n − λ k ) {\displaystyle \left(x^{n}-\lambda...