Conflicting Nomenclatures and Notations
Nomenclatures differ. Some statisticians, following the probabilist William Feller, call this matrix the variance of the random vector, because it is the natural generalization to higher dimensions of the 1-dimensional variance. Others call it the covariance matrix, because it is the matrix of covariances between the scalar components of the vector . Thus
![\operatorname{var}(\textbf{X})
=
\operatorname{cov}(\textbf{X})
=
\mathrm{E}
\left[ (\textbf{X} - \mathrm{E} ) (\textbf{X} - \mathrm{E} )^{\rm T}
\right].](http://upload.wikimedia.org/math/f/2/0/f2051a015d45704a69e67b07a5de39f1.png)
However, the notation for the cross-covariance between two vectors is standard:
![\operatorname{cov}(\textbf{X},\textbf{Y})
=
\mathrm{E}
\left[ (\textbf{X} - \mathrm{E}) (\textbf{Y} - \mathrm{E})^{\rm T}
\right].](http://upload.wikimedia.org/math/2/d/2/2d2205c51ce9e4c73d79795c8db6b49d.png)
The var notation is found in William Feller's two-volume book An Introduction to Probability Theory and Its Applications, but both forms are quite standard and there is no ambiguity between them.
The matrix is also often called the variance-covariance matrix since the diagonal terms are in fact variances.
Read more about this topic: Covariance Matrix
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