Covariance
Cross-covariance matrix of real random vectors

For real random vectors X ∈ R m {\displaystyle \mathbf {X} \in \mathbb {R} ^{m}} and Y ∈ R n {\displaystyle \mathbf {Y} \in \mathbb {R} ^{n}}, the m × n {\displaystyle m\times n} cross-covariance matrix is equal to

where Y T {\displaystyle \mathbf {Y} ^{\mathrm {T} }} is the transpose of the vector (or matrix) Y {\displaystyle \mathbf {Y} }.

The (i, j) {\displaystyle (i,j)} -th element of this matrix is equal to the covariance cov (X i, Y j) {\displaystyle \operatorname {cov} (X_{i},Y_{j})} between the i -th scalar component of X {\displaystyle \mathbf {X} } and the j -th scalar component of Y {\displaystyle \mathbf {Y} }. In particular, cov (Y, X) {\displaystyle \operatorname {cov} (\mathbf {Y},\mathbf {X})} is the transpose of cov (X, Y) {\displaystyle \operatorname {cov} (\mathbf {X},\mathbf {Y})}.
punctuation:no_comma, length_constraints:number_paragraphs[6], detectable_format:multiple_sections[3]