polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

''P'' is ''annihilating'' or called an annihilating polynomial in

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathemat ...

and

operator theory In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operato ...

if the polynomial considered as a function of the

linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...

or a matrix ''A'' evaluates to zero, i.e., is such that ''P(A) = 0''. Note that all

characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The ...

s and minimal polynomials of ''A'' are annihilating polynomials. In fact, every annihilating polynomial is the multiple of the minimal polynomial of an operator ''A''.Hoffman, K., Kunze, R., "Linear Algebra", 2nd ed., 1971, Prentice-Hall. (Definition on page 191 of section 6.3)

References

{{reflist Matrix theory Polynomials Linear algebra

See also

References