Editing Covariance matrix (section)

===Relation to the correlation matrix===
{{further|Correlation matrix}}
An entity closely related to the covariance matrix is the matrix of [[Pearson product-moment correlation coefficient]]s between each of the random variables in the random vector <math>\mathbf{X}</math>, which can be written as
<math display="block">\operatorname{corr}(\mathbf{X}) = \big(\operatorname{diag}(\operatorname{K}_{\mathbf{X}\mathbf{X}})\big)^{-\frac{1}{2}} \, \operatorname{K}_{\mathbf{X}\mathbf{X}} \, \big(\operatorname{diag}(\operatorname{K}_{\mathbf{X}\mathbf{X}})\big)^{-\frac{1}{2}},</math>
where  <math>\operatorname{diag}(\operatorname{K}_{\mathbf{X}\mathbf{X}})</math> is the matrix of the diagonal elements of <math>\operatorname{K}_{\mathbf{X}\mathbf{X}}</math> (i.e., a [[diagonal matrix]] of the variances of <math>X_i</math>  for <math>i = 1, \dots, n</math>).

Equivalently, the correlation matrix can be seen as the covariance matrix of the [[standardized variable|standardized random variables]] <math>X_i/\sigma(X_i)</math> for <math>i = 1, \dots, n</math>.
<math display="block">
\operatorname{corr}(\mathbf{X})
= \begin{bmatrix}
 1 & \frac{\operatorname{E}[(X_1 - \mu_1)(X_2 - \mu_2)]}{\sigma(X_1)\sigma(X_2)} & \cdots & \frac{\operatorname{E}[(X_1 - \mu_1)(X_n - \mu_n)]}{\sigma(X_1)\sigma(X_n)} \\ \\
 \frac{\operatorname{E}[(X_2 - \mu_2)(X_1 - \mu_1)]}{\sigma(X_2)\sigma(X_1)} & 1 & \cdots & \frac{\operatorname{E}[(X_2 - \mu_2)(X_n - \mu_n)]}{\sigma(X_2)\sigma(X_n)} \\ \\
 \vdots & \vdots & \ddots & \vdots \\ \\
 \frac{\operatorname{E}[(X_n - \mu_n)(X_1 - \mu_1)]}{\sigma(X_n)\sigma(X_1)} & \frac{\operatorname{E}[(X_n - \mu_n)(X_2 - \mu_2)]}{\sigma(X_n)\sigma(X_2)} & \cdots & 1
\end{bmatrix}.
</math>

Each element on the principal diagonal of a correlation matrix is the correlation of a random variable with itself, which always equals 1. Each [[off-diagonal element]] is between −1 and +1 inclusive.