Minimal polynomial
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.
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.
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In
field theory, a
minimal polynomial is a
polynomial m(
x) in the field
Zp (with
p prime), such that, if we have the field
F=
Zp(α), it is the polynomial of least
degree with
m(α)=0.
The minimal polynomial is unique, and if we have some irreducible polynomial f(x) with f(α)=0, then f is the minimal polynomial of α.