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Correlation matrix

From Encyclopedia of Mathematics
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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
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article