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*.

*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

**Z**

_{p}(with

*p*prime), such that, if we have the field

**F**=

**Z**

_{p}(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.