Knowlpedia

Definition

Averaged second-moment matrix

The average of the outer product of a square-integrable vector with itself.

On a probability measure space, let w∈L2(μ;Rd)w\in L^2(\mu;\mathbb R^d). Its averaged second-moment matrix is

R(w)=∫w⊗w dμ,R(w)ij=∫wiwj dμ.R(w)=\int w\otimes w\,d\mu,\qquad R(w)_{ij}=\int w_iw_j\,d\mu.

Each entry is integrable by Hölder's inequality. It is symmetric and , since ξTR(w)ξ=∫(ξ⋅w)2 dμ≥0\xi^TR(w)\xi=\int(\xi\cdot w)^2\,d\mu\ge0.

Centering and signs

This is an uncentered second moment. If wˉ=∫w dμ\bar w=\int w\,d\mu, its centered version is R(w)−wˉ⊗wˉR(w)-\bar w\otimes\bar w, the covariance matrix. Off-diagonal entries of either matrix can be negative even though the whole matrix is positive semidefinite. An average over a periodic variable is one example of a probability average; no physical randomness is required.