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
y = β 0 + β 1 x 1 + β 2 x 2 + … β k x k + u {\displaystyle y=\beta _{0}+\beta _{1}x_{1}+\beta _{2}x_{2}+\dots \beta _{k}x_{k}+u}
where y {\displaystyle y} is the dependent variable, x 1 , x 2 … , x k {\displaystyle x_{1},x_{2}\dots ,x_{k}} are the independent variables, u {\displaystyle u} is the error, and β 0 , β 1 … , β k {\displaystyle \beta _{0},\beta _{1}\dots ,\beta _{k}} are unknown coefficients to be estimated. Given observations { y i , x i 1 , x i 2 , … , x i k } i = 1 n {\displaystyle \left\{y_{i},x_{i1},x_{i2},\dots ,x_{ik}\right\}_{i=1}^{n}} , we have a system of n {\displaystyle n} linear equations that can be expressed in matrix notation.
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