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The matrix formed from the pairwise covariances of several random variables; more precisely, for the  $  k $-
dimensional vector  $  X = ( X _ {1} \dots X _ {k} ) $
the covariance matrix is the square matrix  $  \Sigma = {\mathsf E} [ ( X - {\mathsf E} X ) ( X - {\mathsf E} X )  ^ {T} ] $, 
where  $  {\mathsf E} X = ( {\mathsf E} X _ {1} \dots {\mathsf E} X _ {k} ) $
is the vector of mean values. The components of the covariance matrix are:

$$ 
\sigma _ {ij}  =  {\mathsf E}
[ ( X _ {i} - {\mathsf E} X _ {i} ) ( X _ {j} - {\mathsf E} X _ {j} ) ]
 = \ 
 \mathop{\rm cov} ( X _ {i} , X _ {j} ) ,
$$

$$ 
i , j  =  1 \dots k ,
$$

and for  $  i = j $
they are the same as  $  {\mathsf D} X _ {i} $(
$  =  \mathop{\rm var} ( X _ {i} ) $) 
(that is, the variances of the  $  X _ {i} $
lie on the principal diagonal). The covariance matrix is a symmetric positive semi-definite matrix. If the covariance matrix is positive definite, then the distribution of  $  X $
is non-degenerate; otherwise it is degenerate. For the random vector  $  X $
the covariance matrix plays the same role as the variance of a random variable. If the variances of the random variables  $  X _ {1} \dots X _ {k} $
are all equal to 1, then the covariance matrix of  $  X = ( X _ {1} \dots X _ {k} ) $
is the same as the [[Correlation matrix|correlation matrix]].

The sample covariance matrix for the sample  $  X  ^ {(1)} \dots X  ^ {(n)} $, 
where the  $  X  ^ {(m)} $, 
$  m = 1 \dots n $, 
are independent and identically-distributed random  $  k $-
dimensional vectors, consists of the variance and covariance estimators:

$$ 
S  =  
\frac{1}{n-1}\sum_{m=1}^n
( X  ^ {(m)} - \overline{X}\; ) ( X  ^ {(m)} - \overline{X}\; )  ^ {T} ,
$$

where the vector  $  \overline{X}\; $
is the arithmetic mean of the  $  X  ^ {(1)} \dots X  ^ {(n)} $. 
If the  $  X  ^ {(1)} \dots X  ^ {(n)} $
are multivariate normally distributed with covariance matrix  $  \Sigma $, 
then  $  S ( n - 1 ) / n $
is the maximum-likelihood estimator of  $  \Sigma $; 
in this case the joint distribution of the elements of the matrix  $  ( n - 1 ) S $
is called the [[Wishart distribution]]; it is one of the fundamental distributions in multivariate statistical analysis by means of which hypotheses concerning the covariance matrix  $  \Sigma $
can be tested.