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A new method is presented for estimating the power spectral density matrix for multichannel data, given correlation values for linear combinations of the channels and given an initial estimate of the spectral density matrix. A derivation of the method from the relative-entropy principle is given. The results differ significantly from the Multisignal Relative-Entropy ("Minimum-Cross-Entropy") Spectrum Analysis of Johnson and Shore because the present method does not arbitrarily force the final distributions of the various channels to be independent. For the special case when correlation values are given only for the sum of the channels, Multichannel Relative-Entropy Spectrum Analysis is shown to reduce to a two-stage procedure: first a smooth power-spectrum model is fitted to the correlations of the sum; then final estimates of the spectra and cross spectra are obtained through linear filtering. Illustrative numerical examples are presented.