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

d{bmatrix}}={\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\end{bmatrix}}} Annihilating polynomial Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol...

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...

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

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

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...

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

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

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...

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...

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...

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...

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...

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

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

{\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...

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...

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...

_{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...

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...