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 optimization, since positive semidefinite moment matrices correspond to polynomials which are sums of squares, and econometrics.

A multiple linear regression model can be written as

where ${\displaystyle y}$ is the explained variable, ${\displaystyle x_{1},x_{2}\dots x_{k}}$ are the explanatory variables, ${\displaystyle u}$ is the error, and ${\displaystyle \beta _{0},\beta _{1}\dots \beta _{k}}$ are unknown coefficients to be estimated. Given observations ${\displaystyle \left\{y_{i},x_{1i},x_{2i},\dots x_{ki}\right\}_{i=1}^{n}}$, we have a system of ${\displaystyle n}$ linear equations that can be expressed in matrix notation.

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