Generalized variance is the determinant of correlation matrix. Does increasing the off-diagonal entries (correlation coefficients) decreases the determinant? Is a proof available? All elements are positive. Can we deduce from Hadamard inequality of determinant?
2026-04-01 08:00:45.1775030445
Generalized variance
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It can do either. $\;$ Suppose the correlation matrix is $\begin{bmatrix} 1 & x \\ x & 1 \end{bmatrix}$.
$\operatorname{det}\left(\begin{bmatrix} 1 & x \\ x & 1 \end{bmatrix}\right) = 1\cdot 1-x\cdot x = 1-x^2$
If $x<0$ then increasing the off-diagonal entries increases the determinant.
If $0<x$ then increasing the off-diagonal entires decreases the determinant.