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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 the exponent beyond 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".
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• Minimal_polynomial_(linear_algebra) wikiPageOutDegree "28".
• Minimal_polynomial_(linear_algebra) wikiPageRevisionID "701111727".
• Minimal_polynomial_(linear_algebra) wikiPageWikiLink Category:Matrix_theory.
• Minimal_polynomial_(linear_algebra) wikiPageWikiLink Category:Polynomials.
• Minimal_polynomial_(linear_algebra) wikiPageWikiLink Cayley–Hamilton_theorem.
• Minimal_polynomial_(linear_algebra) wikiPageWikiLink Characteristic_(algebra).
• Minimal_polynomial_(linear_algebra) wikiPageWikiLink Characteristic_polynomial.
• Minimal_polynomial_(linear_algebra) wikiPageWikiLink Codimension.
• Minimal_polynomial_(linear_algebra) wikiPageWikiLink Diagonalizable_matrix.
• Minimal_polynomial_(linear_algebra) wikiPageWikiLink Eigenvalues_and_eigenvectors.
• Minimal_polynomial_(linear_algebra) wikiPageWikiLink Endomorphism.
• Minimal_polynomial_(linear_algebra) wikiPageWikiLink Field_(mathematics).
• Minimal_polynomial_(linear_algebra) wikiPageWikiLink Generalized_eigenvector.
• Minimal_polynomial_(linear_algebra) wikiPageWikiLink Ideal_(ring_theory).
• Minimal_polynomial_(linear_algebra) wikiPageWikiLink Involution_(mathematics).
• Minimal_polynomial_(linear_algebra) wikiPageWikiLink Irreducible_polynomial.
• Minimal_polynomial_(linear_algebra) wikiPageWikiLink Linear_algebra.
• Minimal_polynomial_(linear_algebra) wikiPageWikiLink Linear_independence.
• Minimal_polynomial_(linear_algebra) wikiPageWikiLink Matrix_(mathematics).
• Minimal_polynomial_(linear_algebra) wikiPageWikiLink Monic_polynomial.
• Minimal_polynomial_(linear_algebra) wikiPageWikiLink Principal_ideal_domain.
• Minimal_polynomial_(linear_algebra) wikiPageWikiLink Projection_(linear_algebra).
• Minimal_polynomial_(linear_algebra) wikiPageWikiLink Representation_theory.
• Minimal_polynomial_(linear_algebra) wikiPageWikiLink Vector_space.
• Minimal_polynomial_(linear_algebra) wikiPageWikiLinkText ":Minimal polynomial (linear algebra)".
• Minimal_polynomial_(linear_algebra) wikiPageWikiLinkText ":Minimal_polynomial_(linear_algebra)".
• Minimal_polynomial_(linear_algebra) wikiPageWikiLinkText "Minimal polynomial (linear algebra)".
• Minimal_polynomial_(linear_algebra) wikiPageWikiLinkText "Minimal polynomial".
• Minimal_polynomial_(linear_algebra) wikiPageWikiLinkText "minimal polynomial".
• Minimal_polynomial_(linear_algebra) wikiPageWikiLinkText "minimal".
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• Minimal_polynomial_(linear_algebra) subject Category:Matrix_theory.
• Minimal_polynomial_(linear_algebra) subject Category:Polynomials.
• Minimal_polynomial_(linear_algebra) hypernym P.
• Minimal_polynomial_(linear_algebra) type Album.
• Minimal_polynomial_(linear_algebra) type Type.
• Minimal_polynomial_(linear_algebra) type Function.
• Minimal_polynomial_(linear_algebra) type Polynomial.
• Minimal_polynomial_(linear_algebra) type Redirect.
• Minimal_polynomial_(linear_algebra) type Type.
• 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)".
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• Minimal_polynomial_(linear_algebra) sameAs 最小多項式_(線型代数学).
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• Minimal_polynomial_(linear_algebra) wasDerivedFrom Minimal_polynomial_(linear_algebra)?oldid=701111727.
• Minimal_polynomial_(linear_algebra) isPrimaryTopicOf Minimal_polynomial_(linear_algebra).