Monday, March 20, 2006

Determinant of Variance-Covariance Matrix


Of great interest in statistics is the determinant of a square symmetric matrix D whose diagonal elements are sample variances and whose off-diagonal elements are sample covariances. Symmetry means that the matrix and its transpose are identical (i.e., A = A'). An example is
D = [s1**2 s1*s2*r12 ... s1*sp*r1p; s2*s1*r21 s2**2 ... s2*sp*r2p; .... ; sp*s1*rp1 sp*s2*rp2 ... sp**2]

where s1 and s2 are sample standard deviations and rij is the sample correlation.

D is the sample variance-covariance matrix for observations of a multivariate vector of p elements. The determinant of D, in this case, is sometimes called the generalized variance.

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