Variance-Covariance Matrix of Random Vectors by Qiang Gao, updated at May 8, 2017 The variance-covariance matrix of a random vector $$ \mathbf{x} $$ is defined as Var ( x ) ≡ E [ ( x − E ( x ) ) ( x − E ( x ) ) ′ ] (the definition) = E [ x x ′ − x E ( x ) ′ − E ( x ) x ′ + E ( x ) E ( x ) ′ ] = E ( x x ′ ) − E ( x ) E ( x ) ′ − E ( x ) E ( x ) ′ + E ( x ) E ( x ) ′ = E ( x x ′ ) − E ( x ) E ( x ) ′ . (the formula) The last equation is the convenient formula for calculating variance. The covariance matrix between two random vectors $$ \mathbf{x} $$ and $$ \mathbf{y} $$ is defined as Cov ( x , y ) ≡ E [ ( x − E ( x ) ) ( y − E ( y ) ) ′ ] (the definition) = E [ x y ′ − x E ( y ) ′ − E ( x ) y ′ + E ( x ) E ( y ) ′ ] = E ( x y ′ ) − E ( x ) E ( y ) ′ − E ( x ) E ( y ) ′ + E ( x ) E ( y ) ′ = E ( x x ′ ) − E ( x ) E ( x ) ′ . (the formula) The last equation is the convenient formula for calculating variance. Copyright ©2017 by Qiang Gao
