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What is the difference between the maximum value of cross-correlation value of RVs X, Y and maximum eigenvalue of Covariance matrix of these same RVs X and Y? Are both same and just represents the correlation between two RVs?

Is maximum Pearson correlation between two data vectors will be equal to the maximum eigenvalue of the Covariance/correlation matrix of that two data vectors?

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    $\begingroup$ You are mixing two related but not equivalent concepts: covariance is absolute where correlation (for instance Pearson correlation) is relative (scaled). Regarding your question: Pearson coefficient is not the same as any eigen value of the covariance matrix: take the case where both sets are perfectly uncorrelated ($\rho = 0$): the covariance matrix is diagonal with coefficients equal to the variance in each set, so the maximum eigen value is not 0. Pearson coefficient is linked to the covariance matrix by: $\rho = \frac{cov(X,Y)}{\sigma_X\cdot\sigma_Y} = K_{12} / \sqrt{K_{11}\cdot K_{22}}$ $\endgroup$ – Romain Reboulleau Dec 12 '18 at 19:58
  • $\begingroup$ If "RV" means random variable, then the question is strange because random variables don't have any "cross-correlation." You seem to be confusing them with stochastic processes, time series, vectors, or signals. Two parallel data vectors do indeed have a Pearson correlation, so technically "maximum Pearson correlation" makes sense as the maximum of a set of one number, but this suggests you're not really thinking of the Pearson correlation as it's usually understood. Could you edit your question to clarify these points? $\endgroup$ – whuber Dec 12 '18 at 21:14

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