Editing Covariance matrix (section)

==Covariance matrix as a linear operator==
{{main|Covariance operator}}
Applied to one vector, the covariance matrix maps a linear combination '''c''' of the random variables '''X''' onto a vector of covariances with those variables: <math>\mathbf c^\mathsf{T} \Sigma = \operatorname{cov}(\mathbf c^\mathsf{T} \mathbf X, \mathbf X)</math>. Treated as a [[bilinear form]], it yields the covariance between the two linear combinations: <math>\mathbf d^\mathsf{T} \boldsymbol\Sigma \mathbf c = \operatorname{cov}(\mathbf d^\mathsf{T} \mathbf X, \mathbf c^\mathsf{T} \mathbf X)</math>. The variance of a linear combination is then <math>\mathbf c^\mathsf{T} \boldsymbol\Sigma \mathbf c</math>, its covariance with itself.

Similarly, the (pseudo-)inverse covariance matrix provides an inner product <math>\langle c - \mu| \Sigma^+ |c - \mu\rangle</math>, which induces the [[Mahalanobis distance]], a measure of the "unlikelihood" of ''c''.{{Citation needed|date=February 2012}}