# Correlation matrix

The matrix of correlation coefficients of several random variables. If $ X _ {1} \dots X _ {n} $
are random variables with non-zero variances $ \sigma _ {1} ^ {2} \dots \sigma _ {n} ^ {2} $,
then the matrix entries $ \rho _ {ij} $(
$ i \neq j $)
are equal to the correlation coefficients (cf. Correlation coefficient) $ \rho ( X _ {i} , X _ {j} ) $;
for $ i = j $
the element is defined to be 1. The properties of the correlation matrix $ {\mathsf P} $
are determined by the properties of the covariance matrix $ \Sigma $,
by virtue of the relation $ \Sigma = B {\mathsf P} B $,
where $ B $
is the diagonal matrix with (diagonal) entries $ \sigma _ {1} \dots \sigma _ {n} $.

**How to Cite This Entry:**

Correlation matrix.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Correlation_matrix&oldid=46526