Minimal Polynomial (linear Algebra)

Minimal Polynomial (linear Algebra)

In linear algebra, the minimal polynomial μ_A of an n-by-n matrix A over a field F is the monic polynomial P over F of least degree such that P(A)=0. Any other polynomial Q with Q(A) = 0 is a (polynomial) multiple of μ_A.

The following three statements are equivalent:

1. λ is a root of μ_A,
2. λ is a root of the characteristic polynomial χ_A of A,
3. λ is an eigenvalue of matrix A.

The multiplicity of a root λ of μ_A is the largest power m such that Ker((A − λI_n)m) strictly contains Ker((A − λI_n)m−1) (increasing the exponent up to m will give ever larger kernels, but further increasing m will just give the same kernel).

If the field F is not algebraically closed, then the minimal and characteristic polynomials need not factor according to their roots (in F) alone, in other words they may have irreducible polynomial factors of degree greater than 1. For irreducible polynomials P one has similar equivalences:

1. P divides μ_A,
2. P divides χ_A,
3. the kernel of P(A) has dimension at least 1.
4. the kernel of P(A) has dimension at least deg(P).

Like the characteristic polynomial, the minimal polynomial does not depend on the base field, in other words considering the matrix as one with coefficients in a larger field does not change the minimal polynomial. The reason is somewhat different than for the characteristic polynomial (where it is immediate from the definition of determinants), namely the fact that the minimal polynomial is determined by the relations of linear dependence between the powers of A: extending the base field will not introduce any new such relations (nor of course will it remove existing ones).

The minimal polynomial is often the same as the characteristic polynomial, but not always. For example, if A is a multiple of the identity matrix, then its minimal polynomial is X − a since the kernel of is already the entire space; on the other hand its characteristic polynomial is (X − a)n (the only eigenvalue is a, and the degree of the characteristic polynomial is always equal to the dimension of the space). The minimal polynomial always divides the characteristic polynomial, which is one way of formulating the Cayley–Hamilton theorem (for the case of matrices over a field).