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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 non-zero polynomial f with p(A) = 0 is a 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 geometrical multiplicity of λ and is the size of the largest Jordan block corresponding to λ.
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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