Multiplication Matrix

A multiplication matrix of an element in a finite-dimensional associative commutative algebra with multiplicative identity over a field is the matrix, in a chosen vector basis, of the -linear transformation

(1)

Changing the vector basis replaces by a similar matrix. For elements , the matrices satisfy

(2)

For polynomial equations, let , where is a zero-dimensional ideal. Choose polynomial representatives , ..., of a vector basis of this quotient ring. The th column of is determined by

(3)

A Gröbner basis computes these expressions by polynomial division when the are the monomials not divisible by any of its leading monomials (Sturmfels 2002, p. 17).

When , let be the finite set of common zeros of . For , the evaluation vector satisfies

(4)

Thus the eigenvalues of are the values . More precisely, its characteristic polynomial is

(5)

where is the identity matrix and is the dimension over of the localization of at the maximal ideal corresponding to (Cox 2021). Common left eigenvectors of , ..., give consistent coordinate tuples. Separate lists of eigenvalues do not in general determine which coordinates belong to the same solution. If , where is the ideal radical, these matrices can be diagonalized in a common vector basis (Sturmfels 2002, pp. 17-18). This gives an eigenvalue method for solving finite systems of polynomial equations (Auzinger and Stetter 1988, Trott 2006).

In one variable, multiplication by on , for a monic polynomial of polynomial degree , gives the standard companion matrix in the vector basis , , ..., .