About: Minimal polynomial (linear algebra)

Facets (new session)
Description
Metadata
Settings
- Rule:
- Inverse Functional Properties:
- "Same As":

About: Minimal polynomial (linear algebra) Goto Sponge NotDistinct Permalink

An Entity of Type : dbo:Album, within Data Space : el.dbpedia.org associated with source document(s)

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: 1. * λ is a root of μA, 2. * λ is a root of the characteristic polynomial χA of A, 3. * λ is an eigenvalue of matrix A. 1. * P divides μA, 2. * P divides χA, 3. * the kernel of P(A) has dimension at least 1. 4. * the kernel of P(A) has dimension at least deg(P).

Attributes	Values
rdf:type	album
rdfs:label	Minimal polynomial (linear algebra) (en)
rdfs: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: 1. * λ is a root of μA, 2. * λ is a root of the characteristic polynomial χA of A, 3. * λ is an eigenvalue of matrix A. 1. * P divides μA, 2. * P divides χA, 3. * the kernel of P(A) has dimension at least 1. 4. * the kernel of P(A) has dimension at least deg(P). (en)
sameAs	Minimal polynomial (linear algebra) Minimal polynomial (linear algebra)
dbp:wikiPageUsesTemplate	dbt:= dbt:For dbt:Math dbt:Mvar dbt:Reflist dbt:Unordered_list dbt:Lang_Algebra
Subject	Polynomials Matrix theory
gold:hypernym	P
prov:wasDerivedFrom	http://en.wikipedia.org/wiki/Minimal_polynomial_(linear_algebra)?oldid=1069735119&ns=0
Wikipage page ID	9667107 (xsd:integer)
page length (characters) of wiki page	11120 (xsd:nonNegativeInteger)
Wikipage revision ID	1069735119 (xsd:integer)
Link from a Wikipage to another Wikipage	Logical equivalence Projection (linear algebra) Unit (ring theory) Algebraically closed field Basis (linear algebra) Canonical basis Polynomials Cayley–Hamilton theorem Generalized eigenvector Jordan matrix Matrix theory Dimension (vector space) Zero matrix Zero of a function Characteristic (algebra) Characteristic polynomial Cyclic group Degree of a polynomial Linear algebra Linear independence Linear span Linear subspace Principal ideal domain Codimension Complex number Eigenvalues and eigenvectors Endomorphism Ideal (ring theory) Unit matrix If and only if Kernel (linear algebra) Determinant Diagonalizable matrix Field (mathematics) Involution (mathematics) Irreducible polynomial Monic polynomial Vector space Mathematical proof Matrix (mathematics) History of numbers Nilpotent Polynomial Representation theory
has 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: 1. * λ is a root of μA, 2. * λ is a root of the characteristic polynomial χA of A, 3. * 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: 1. * P divides μA, 2. * P divides χA, 3. * the kernel of P(A) has dimension at least 1. 4. * 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). (en)
foaf:isPrimaryTopicOf	http://en.wikipedia.org/wiki/Minimal_polynomial_(linear_algebra)
is Wikipage redirect of	Minimal polynomial (Linear Algebra) Existence of the minimal polynomial Minimum polynomial Algebraic number minimal polynomial
is Link from a Wikipage to another Wikipage of	Adjacency matrix Arnoldi iteration Block Wiedemann algorithm Cayley–Hamilton theorem Jordan matrix Jordan normal form Krylov subspace Characteristic polynomial List of named matrices List of polynomial topics Companion matrix Contraction (operator theory) Minimal polynomial (Linear Algebra) Frobenius normal form Frobenius theorem (real division algebras) Functional calculus Diagonalizable matrix Expm Matrix polynomial Matrix similarity Multiset Nilpotent matrix Existence of the minimal polynomial Minimum polynomial Algebraic number minimal polynomial
is foaf:primaryTopic of	http://en.wikipedia.org/wiki/Minimal_polynomial_(linear_algebra)

Faceted Search & Find service v1.17_git151 as of Feb 20 2025

Alternative Linked Data Documents: ODE Content Formats:

RDF

ODATA

Microdata

About

OpenLink Virtuoso version 07.20.3240 as of Nov 11 2024, on Linux (x86_64-ubuntu_focal-linux-gnu), Single-Server Edition (71 GB total memory, 1 GB memory in use)
Data on this page belongs to its respective rights holders.
Virtuoso Faceted Browser Copyright © 2009-2025 OpenLink Software