Thomas Lumley • 3/6/2023

Determinant of correlation matrix

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The article presents a theorem and proof demonstrating that the determinant of any correlation matrix cannot exceed 1. The proof leverages the fact that the trace of the matrix equals its dimension p, which is the sum of its eigenvalues. Applying the arithmetic mean-geometric mean (AM-GM) inequality to the eigenvalues shows their geometric mean is ≤1, and thus its p-th power (the determinant) is also ≤1.

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#linear algebra #matrices #determinant
Determinant of correlation matrix

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