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In linear algebra, the minimal polynomial μ A 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 Q(A) = 0 is a (polynomial) multiple of μ A. The following three statements are equivalent: λ is a root of μ A, λ is a root of the characteristic polynomial χ A ...
The minimal polynomial f of α is irreducible, i.e. it cannot be factorized as f = gh for two polynomials g and h of strictly lower degree. To prove this, first observe that any factorization f = gh implies that either g ( α ) = 0 or h ( α ) = 0, because f ( α ) = 0 and F is a field (hence also an integral domain ).
8 Notes. Toggle the table of contents. ... In linear algebra, ... coincides with the minimal polynomial of A, i.e. the minimal ...
The minimal polynomial over K of θ is thus the monic polynomial of minimal degree that has θ as a root. Because L is a field, this minimal polynomial is necessarily irreducible over K. For example, the minimal polynomial (over the reals as well as over the rationals) of the complex number i is +. The cyclotomic polynomials are the minimal ...
A polynomial P is annihilating or called an annihilating polynomial in linear algebra and operator theory if the polynomial considered as a function of the linear operator or a matrix A evaluates to zero, i.e., is such that P(A) = 0. Note that all characteristic polynomials and minimal polynomials of A are annihilating polynomials
A minimal algebra is a finite algebra with more than one element, in which every non-constant unary polynomial is a permutation on its domain. In simpler terms, it’s an algebraic structure where unary operations (those involving a single input) behave like permutations (bijective mappings). These algebras provide intriguing connections ...
Moreover, over a perfect field, a polynomial is separable if and only if it is square-free. Therefore an operator is potentially diagonalisable if and only if its minimal polynomial is square-free. In general (over any field), the minimal polynomial of a linear operator is square-free if and only if the operator is semisimple. [2] (In ...
The Conway polynomial C p,n is defined as the lexicographically minimal monic primitive polynomial of degree n over F p that is compatible with C p,m for all m dividing n.This is an inductive definition on n: the base case is C p,1 (x) = x − α where α is the lexicographically minimal primitive element of F p.
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