Quadratic form (statistics)

In multivariate statistics, if ε {\displaystyle \varepsilon } is a vector of n {\displaystyle n} random variables, and Λ {\displaystyle \Lambda } is an n {\displaystyle n} -dimensional symmetric matrix, then the scalar quantity ε T Λ ε {\displaystyle \varepsilon ^{T}\Lambda \varepsilon } is known as a quadratic form in ε {\displaystyle \varepsilon } . == Expectation == It can be shown that E ⁡ [ ε T Λ ε ] = tr ⁡ [ Λ Σ ] + μ T Λ μ {\displaystyle \operatorname {E} \left[\varepsilon ^{T}\Lambda \varepsilon \right]=\operatorname {tr} \left[\Lambda \Sigma \right]+\mu ^{T}\Lambda \mu } where μ {\displaystyle \mu } and Σ {\displaystyle \Sigma } are the expected value and variance-covariance matrix of ε {\displaystyle \varepsilon } , respectively, and tr denotes the trace of a matrix.

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