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Revision as of 03:21, 14 October 2005 editDysprosia (talk | contribs)28,388 editsm it's better to have a nontrivial example← Previous edit Revision as of 03:53, 14 October 2005 edit undoKusma (talk | contribs)Autopatrolled, Administrators59,515 edits the point of my edit was the 4 in the exponent, which is correct iff n=4Next edit →
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The multiplicity of a root λ of ''p''(''x'') is the ''geometric multiplicity'' of λ and is the size of the largest ] corresponding to λ and the dimension of the corresponding eigenspace. The multiplicity of a root λ of ''p''(''x'') is the ''geometric multiplicity'' of λ and is the size of the largest ] corresponding to λ and the dimension of the corresponding eigenspace.


The minimum polynomial is not always the same as the characteristic polynomial. Consider the matrix <math>4I</math>, it has characteristic polynomial <math>(x-4)^4</math>, but <math>(x-4)</math> is the minimum polynomial, as <math>4I-4I=0</math> as desired. That the minimal polynomial always divides the characteristic polynomial is a consequence of the ]. The minimal polynomial is not always the same as the characteristic polynomial. Consider the matrix <math>4I</math>, it 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 ].


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Revision as of 03:53, 14 October 2005

The minimal polynomial of an n-by-n matrix A over a field F is the polynomial p(x) with leading coefficient 1 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:

  1. λ∈F is a root of p(x),
  2. λ is a root of the characteristic polynomial of A,
  3. λ is an eigenvalue of A.

The multiplicity of a root λ of p(x) is the geometric multiplicity of λ and is the size of the largest Jordan block corresponding to λ and the dimension of the corresponding eigenspace.

The minimal polynomial is not always the same as the characteristic polynomial. Consider the matrix 4 I {\displaystyle 4I} , it has characteristic polynomial ( x − 4 ) n {\displaystyle (x-4)^{n}} . However, the minimal polynomial is x − 4 {\displaystyle x-4} , since 4 I − 4 I = 0 {\displaystyle 4I-4I=0} as desired, so they are different for n ≥ 2 {\displaystyle n\geq 2} . That the minimal polynomial always divides the characteristic polynomial is a consequence of the Cayley–Hamilton theorem.

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.

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