View source for Correlation matrix

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" />.