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  2. Monic polynomial - Wikipedia

    en.wikipedia.org/wiki/Monic_polynomial

    In this case, a polynomial may be said to be monic, if it has 1 as its leading coefficient (for the monomial order). For every definition, a product of monic polynomials is monic, and, if the coefficients belong to a field, every polynomial is associated to exactly one monic polynomial.

  3. Minimal polynomial (field theory) - Wikipedia

    en.wikipedia.org/wiki/Minimal_polynomial_(field...

    The minimal polynomial f of α is unique.. To prove this, suppose that f and g are monic polynomials in J α of minimal degree n > 0. We have that r := f−g ∈ J α (because the latter is closed under addition/subtraction) and that m := deg(r) < n (because the polynomials are monic of the same degree).

  4. Companion matrix - Wikipedia

    en.wikipedia.org/wiki/Companion_matrix

    The roots of the characteristic polynomial () are the eigenvalues of ().If there are n distinct eigenvalues , …,, then () is diagonalizable as () =, where D is the diagonal matrix and V is the Vandermonde matrix corresponding to the λ 's: = [], = [].

  5. Minimal polynomial (linear algebra) - Wikipedia

    en.wikipedia.org/wiki/Minimal_polynomial_(linear...

    In linear algebra, the minimal polynomial μ A of an n × 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: λ is a root of μ A, λ is a root of the characteristic polynomial χ A ...

  6. Characteristic polynomial - Wikipedia

    en.wikipedia.org/wiki/Characteristic_polynomial

    That polynomial differs from the one defined here by a sign (), so it makes no difference for properties like having as roots the eigenvalues of ; however the definition above always gives a monic polynomial, whereas the alternative definition is monic only when is even.

  7. Monic - Wikipedia

    en.wikipedia.org/wiki/Monic

    Monic morphism, a special kind of morphism in category theory Monic polynomial , a polynomial whose leading coefficient is one A synonym for monogenic , which has multiple uses in mathematics

  8. Primitive polynomial (field theory) - Wikipedia

    en.wikipedia.org/wiki/Primitive_polynomial...

    In finite field theory, a branch of mathematics, a primitive polynomial is the minimal polynomial of a primitive element of the finite field GF(p m).This means that a polynomial F(X) of degree m with coefficients in GF(p) = Z/pZ is a primitive polynomial if it is monic and has a root α in GF(p m) such that {,,,,, …} is the entire field GF(p m).

  9. Matrix (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Matrix_(mathematics)

    The polynomial p A in an indeterminate X given by evaluation of the determinant det(X I n − A) is called the characteristic polynomial of A. It is a monic polynomial of degree n. Therefore the polynomial equation p A (λ) = 0 has at most n different solutions, that is, eigenvalues of the matrix. [42] They may be complex even if the entries of ...