# Whitening of a Multivariate Gaussian Random Vector

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Whitening is the process of transforming a random vector with a known covariance matrix into a new random vector with a covariance matrix equal to the identity matrix. Accordingly, the elements of the new random vector have variance 1 and are uncorrelated. If the original random vector is a multidimensional Gaussian, then the elements of the new random vector have variance 1 and are statistically independent.

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Contributed by: Victor S. Frost (August 27)

(University of Kansas)

Open content licensed under CC BY-NC-SA

## Details

Let be the covariance matrix of a random vector . Let be a diagonal matrix having the eigenvalues of as its diagonal elements. Define such that the columns of are the eigenvectors of the covariance matrix . Then the whitening matrix is . The whitened data is obtained from the samples of the correlated random vector using .

References

[1] S. S. Haykin, *Adaptive Filter Theory*, Englewood Cliffs, NJ: Prentice-Hall, 1986.

[2] K. S. Shanmugan and A. M. Breipohl, *Random Signals: Detection, Estimation and Data Analysis*, New York: Wiley, 1988.

[3] R. W. Picard. "Topic: Decorrelating and Then Whitening Data." (Aug 11, 2023) courses.media.mit.edu/2010fall/mas622j/whiten.pdf.

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