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- Minimal_polynomial_(linear_algebra) abstract "In linear algebra, the 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 Q(A) = 0 is a (polynomial) multiple of μA.The following three statements are equivalent: λ is a root of μA, λ is a root of the characteristic polynomial χA of A, λ is an eigenvalue of matrix A.The multiplicity of a root λ of μA is the largest power m such that Ker((A − λIn)m) strictly contains Ker((A − λIn)m−1). In other words, increasing the exponent up to m will give ever larger kernels, but further increasing m will just give the same kernel.If the field F is not algebraically closed, then the minimal and characteristic polynomials need not factor according to their roots (in F) alone, in other words they may have irreducible polynomial factors of degree greater than 1. For irreducible polynomials P one has similar equivalences: P divides μA, P divides χA, the kernel of P(A) has dimension at least 1. the kernel of P(A) has dimension at least deg(P).Like the characteristic polynomial, the minimal polynomial does not depend on the base field, in other words considering the matrix as one with coefficients in a larger field does not change the minimal polynomial. The reason is somewhat different from for the characteristic polynomial (where it is immediate from the definition of determinants), namely the fact that the minimal polynomial is determined by the relations of linear dependence between the powers of A: extending the base field will not introduce any new such relations (nor of course will it remove existing ones).The minimal polynomial is often the same as the characteristic polynomial, but not always. For example, if A is a multiple aIn of the identity matrix, then its minimal polynomial is X − a since the kernel of aIn − A = 0 is already the entire space; on the other hand its characteristic polynomial is (X − a)n (the only eigenvalue is a, and the degree of the characteristic polynomial is always equal to the dimension of the space). The minimal polynomial always divides the characteristic polynomial, which is one way of formulating the Cayley–Hamilton theorem (for the case of matrices over a field).".
- Minimal_polynomial_(linear_algebra) wikiPageID "9667107".
- Minimal_polynomial_(linear_algebra) wikiPageRevisionID "601888129".
- Minimal_polynomial_(linear_algebra) hasPhotoCollection Minimal_polynomial_(linear_algebra).
- Minimal_polynomial_(linear_algebra) subject Category:Matrix_theory.
- Minimal_polynomial_(linear_algebra) subject Category:Polynomials.
- Minimal_polynomial_(linear_algebra) type Abstraction100002137.
- Minimal_polynomial_(linear_algebra) type Function113783816.
- Minimal_polynomial_(linear_algebra) type MathematicalRelation113783581.
- Minimal_polynomial_(linear_algebra) type Polynomial105861855.
- Minimal_polynomial_(linear_algebra) type Polynomials.
- Minimal_polynomial_(linear_algebra) type Relation100031921.
- Minimal_polynomial_(linear_algebra) comment "In linear algebra, the 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 Q(A) = 0 is a (polynomial) multiple of μA.The following three statements are equivalent: λ is a root of μA, λ is a root of the characteristic polynomial χA of A, λ is an eigenvalue of matrix A.The multiplicity of a root λ of μA is the largest power m such that Ker((A − λIn)m) strictly contains Ker((A − λIn)m−1).".
- Minimal_polynomial_(linear_algebra) label "Minimal polynomial (linear algebra)".
- Minimal_polynomial_(linear_algebra) label "Minimalpolynom".
- Minimal_polynomial_(linear_algebra) label "Polinomio minimo".
- Minimal_polynomial_(linear_algebra) label "Polinomio mínimo".
- Minimal_polynomial_(linear_algebra) label "Polynôme minimal d'un endomorphisme".
- Minimal_polynomial_(linear_algebra) label "Wielomian minimalny".
- Minimal_polynomial_(linear_algebra) label "Минимальный многочлен матрицы".
- Minimal_polynomial_(linear_algebra) label "最小多項式".
- Minimal_polynomial_(linear_algebra) label "極小多項式".
- Minimal_polynomial_(linear_algebra) sameAs Minimální_polynom_(lineární_algebra).
- Minimal_polynomial_(linear_algebra) sameAs Minimalpolynom.
- Minimal_polynomial_(linear_algebra) sameAs Ελάχιστο_πολυώνυμο.
- Minimal_polynomial_(linear_algebra) sameAs Polinomio_mínimo.
- Minimal_polynomial_(linear_algebra) sameAs Polynôme_minimal_d'un_endomorphisme.
- Minimal_polynomial_(linear_algebra) sameAs Polinomio_minimo.
- Minimal_polynomial_(linear_algebra) sameAs 最小多項式.
- Minimal_polynomial_(linear_algebra) sameAs Wielomian_minimalny.
- Minimal_polynomial_(linear_algebra) sameAs m.02pnlcy.
- Minimal_polynomial_(linear_algebra) sameAs Q1163608.
- Minimal_polynomial_(linear_algebra) sameAs Q1163608.
- Minimal_polynomial_(linear_algebra) sameAs Minimal_polynomial_(linear_algebra).
- Minimal_polynomial_(linear_algebra) wasDerivedFrom Minimal_polynomial_(linear_algebra)?oldid=601888129.
- Minimal_polynomial_(linear_algebra) isPrimaryTopicOf Minimal_polynomial_(linear_algebra).