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Difference between revisions of "Correlation matrix"

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The matrix of correlation coefficients of several random variables. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026570/c0265701.png" /> are random variables with non-zero variances <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026570/c0265702.png" />, then the matrix entries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026570/c0265703.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026570/c0265704.png" />) are equal to the correlation coefficients (cf. [[Correlation coefficient|Correlation coefficient]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026570/c0265705.png" />; for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026570/c0265706.png" /> the element is defined to be 1. The properties of the correlation matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026570/c0265707.png" /> are determined by the properties of the [[Covariance matrix|covariance matrix]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026570/c0265708.png" />, by virtue of the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026570/c0265709.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026570/c02657010.png" /> is the diagonal matrix with (diagonal) entries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026570/c02657011.png" />.
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The matrix of correlation coefficients of several random variables. If  $  X _ {1} \dots X _ {n} $
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are random variables with non-zero variances $  \sigma _ {1}  ^ {2} \dots \sigma _ {n}  ^ {2} $,  
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then the matrix entries $  \rho _ {ij} $(
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$  i \neq j $)  
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are equal to the correlation coefficients (cf. [[Correlation coefficient|Correlation coefficient]]) $  \rho ( X _ {i} , X _ {j} ) $;  
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for $  i = j $
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the element is defined to be 1. The properties of the correlation matrix $  {\mathsf P} $
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are determined by the properties of the [[Covariance matrix|covariance matrix]] $  \Sigma $,  
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by virtue of the relation $  \Sigma = B {\mathsf P} B $,  
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where $  B $
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is the diagonal matrix with (diagonal) entries $  \sigma _ {1} \dots \sigma _ {n} $.

Latest revision as of 17:31, 5 June 2020


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