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In the paper, "Data analysis recipes: Fitting a model to data" (Hogg, Bovy, Lang), individual data point variances are found and used for subsequent statistical analysis. The data and corresponding variances are shown in the image below. enter image description here

The paper cites that the full uncertainty covariance matrix for each data point is given by

enter image description here

How is this convariance matrix for each point found?

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This is just the definition of the covariance matrix. Each element along the diagonal is the variance of each random variable. The off-diagonal elements are the covariances, in terms of $\rho_{xy}$ the correlation of coefficient.

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