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...

An 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...

forms", Math. Z. 244, no. 3, 469–477. 2000 (with S. McGarraghy) "Annihilating polynomials, étale algebras, trace forms and the Galois number", Arch. Math...

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

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...

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

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

{\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...

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...

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...

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...

given for polynomial identity rings. The notation Z(R) will be used to denote the center of a ring R. Theorem: If R is a simple polynomial identity ring...

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

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...

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

closure on the polynomial ring is the radical of ideal generated by S. More generally, given a commutative ring R (not necessarily a polynomial ring), there...

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...

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

{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...

of a torsion-free module that is not flat is the ideal (x, y) of the polynomial ring k[x, y] over a field k, interpreted as a module over k[x, y]. Any...

real or more imaginary than the others. Use of neologisms. For example polynomial, homomorphism. Use of symbols as words or phrases. For example, A = B...

{\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...

_{r=0}^{n_{0}}y\cdot (p_{r}(M^{r}x))=0} ; so the minimal polynomial of the matrix annihilates the sequence S {\displaystyle S} and hence S y {\displaystyle...

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...