Minimal polynomial
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In mathematics, the minimal polynomial of an object α is the monic polynomial p of least degree such that p(α)=0. The properties of the minimal polynomial depend on the algebraic structure to which α belongs.
Field theory
In field theory, given a field extension E/F and an element α of E which is algebraic over F, the minimal polynomial of α is the monic polynomial p, with coefficients in F, of least degree such that p(α) = 0. The minimal polynomial is irreducible, and any other non-zero polynomial f with f(α) = 0 is a multiple of p.
Linear algebra
In linear algebra, the minimal polynomial of an n-by-n matrix A over a field F is the monic polynomial p(x) over F of least degree such that p(A)=0. Any other polynomial q with q(A) = 0 is a (polynomial) multiple of p.
The following three statements are equivalent:
- λ∈F is a root of p(x),
- λ is a root of the characteristic polynomial of A,
- λ is an eigenvalue of A.
The multiplicity of a root λ of p(x) is the size of the largest Jordan block corresponding to λ.
The minimal polynomial is not always the same as the characteristic polynomial. Consider the matrix <math>4I_n</math>, which has characteristic polynomial <math>(x-4)^n</math>. However, the minimal polynomial is <math>x-4</math>, since <math>4I-4I=0</math> as desired, so they are different for <math>n\ge 2</math>. That the minimal polynomial always divides the characteristic polynomial is a consequence of the Cayley–Hamilton theorem.ca:Polinomi mínim de:Minimalpolynom es:Polinomio mínimo fr:Polynôme minimal it:Polinomio minimo he:פולינום מינימלי