A polynomial P is annihilating or called an annihilating polynomial in linear algebra and operator theory if the polynomial considered as a function of...
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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...
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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...
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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...
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obtained from the characteristic polynomial of A {\displaystyle A} or, more generally, from any annihilating polynomial of A {\displaystyle A} . This...
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Lag operator (redirect from Lag polynomial)
dividing one such polynomial by another, when each has a finite order (highest exponent), results in an infinite-order polynomial. An annihilator operator, denoted...
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{\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...
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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...
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{\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...
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1007/s00209-003-0490-6. Lewis, D. W.; McGarraghy, S. (2000). "Annihilating polynomials, étale algebras, trace forms and the Galois number". Archiv der...
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extending the field F (whence the "rational"), notably without factoring polynomials, this shows that whether two matrices are similar does not change upon...
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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...
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Jordan normal form (section Characteristic polynomial)
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...
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commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers Z...
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commutative algebra. Prominent examples of commutative rings include polynomial rings, rings of algebraic integers, including the ordinary integers Z...
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characteristic polynomial of A. So the algebraic multiplicity is the multiplicity of the eigenvalue as a zero of the characteristic polynomial. Since any...
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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...
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given binary output sequence. The algorithm will also find the minimal polynomial of a linearly recurrent sequence in an arbitrary field. The field requirement...
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Algebra representation (section Polynomial algebras)
commutatively (which is annihilated by the derived algebra). Thus of central interest are the free commutative algebras, namely the polynomial algebras. In this...
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expanding on the work of Sato and Joseph Bernstein on the Bernstein–Sato polynomial. Early major results were the Kashiwara constructibility theorem and Kashiwara...
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may be computed in terms of division polynomials. Analytic torsion Arithmetic dynamics Flat module Annihilator (ring theory) Localization of a module...
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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...
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solution, one may write the unknown polynomials as polynomials with unknown coefficients. Then, as two polynomials are equal if and only if their coefficients...
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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...
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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...
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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...
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theory itself and for its applications, such as homological properties and polynomial identities. Commutative rings are much better understood than noncommutative...
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{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...
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any. The solutions of homogeneous linear differential equations with polynomial coefficients are called holonomic functions. This class of functions is...
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{\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...
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