Moment matrix

In mathematics, a moment matrix is a special symmetric square matrix whose rows and columns are indexed by monomials. The entries of the matrix depend on the product of the indexing monomials only (cf. Hankel matrices.)
Moment matrices play an important role in polynomial fitting, polynomial optimization (since positive semidefinite moment matrices correspond to polynomials which are sums of squares) and econometrics.
Application in regression.
A multiple linear regression model can be written as
where formula_2 is the explained variable, formula_3 are the explanatory variables, formula_4 is the error, and formula_5 are unknown coefficients to be estimated. Given observations formula_6, we have a system of formula_7 linear equations that can be expressed in matrix notation.
or
where formula_10 and formula_11 are each a vector of dimension formula_12, formula_13 is the design matrix of order formula_14, and formula_15 is a vector of dimension formula_16. Under the Gauss–Markov assumptions, the best linear unbiased estimator of formula_15 is the linear least squares estimator formula_18, involving the two moment matrices formula_19 and formula_20 defined as
and
where formula_19 is a square normal matrix of dimension formula_24, and formula_20 is a vector of dimension formula_26.