Minimal polynomial

The minimal polynomial of an n-by-n matrix A over a field F is the monic polynomial p(x) over F such that p(A)=0.

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 geometrical multiplicity of &lambda and is the size of the largest Jordan block corresponding to &lambda.

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