Editing Covariance matrix (section)

==Estimation==
{{Main|Estimation of covariance matrices}}
If <math>\mathbf{M}_{\mathbf{X}}</math> and <math>\mathbf{M}_{\mathbf{Y}}</math> are centered [[Data matrix (multivariate statistics)|data matrices]] of dimension <math>p \times n</math> and <math>q \times n</math> respectively, i.e. with ''n'' columns of observations of ''p'' and ''q'' rows of variables, from which the row means have been subtracted, then, if the row means were estimated from the data, sample covariance matrices <math>\mathbf{Q}_{\mathbf{XX}}</math> and <math>\mathbf{Q}_{\mathbf{XY}}</math> can be defined to be
<math display="block"> \mathbf{Q}_{\mathbf{XX}} = \frac{1}{n-1} \mathbf{M}_{\mathbf{X}} \mathbf{M}_{\mathbf{X}}^\mathsf{T}, \qquad \mathbf{Q}_{\mathbf{XY}} = \frac{1}{n-1} \mathbf{M}_{\mathbf{X}} \mathbf{M}_{\mathbf{Y}}^\mathsf{T} </math>
or, if the row means were known a priori,
<math display="block"> \mathbf{Q}_{\mathbf{XX}} = \frac{1}{n} \mathbf{M}_{\mathbf{X}} \mathbf{M}_{\mathbf{X}}^\mathsf{T}, \qquad \mathbf{Q}_{\mathbf{XY}} = \frac{1}{n} \mathbf{M}_{\mathbf{X}} \mathbf{M}_{\mathbf{Y}}^\mathsf{T}. </math>

These empirical sample covariance matrices are the most straightforward and most often used estimators for the covariance matrices, but other estimators also exist, including regularised or shrinkage estimators, which may have better properties.