The error matrix, also called the covariance matrix, is the inverse of the second derivative matrix of the (log-likelihood or chisquare) function with respect to its free parameters, usually assumed to be evaluated at the best parameter values (the function minimum). The diagonal elements of the error matrix are the squares of the individual parameter errors, including the effects of correlations with the other parameters.
The inverse of the error matrix, the second derivative matrix, has as diagonal elements the second partial derivatives with respect to one parameter at a time. These diagonal elements are not therefore coupled to any other parameters, but when the matrix is inverted, the diagonal elements of the inverse contain contributions from all the elements of the second derivative matrix, which is ``where the correlations come from''.
Although a parameter may be either positively or negatively correlated with another, the effect of correlations is always to increase the errors on the other parameters in the sense that if a given free parameter suddenly became exactly known (fixed), that would always decrease (or at least not change) the errors on the other parameters. In order to see this effect quantitatively, the following procedure can be used to ``delete'' one parameter from the error matrix, including its effects on the other parameters:
This reduced error matrix will have its diagonal elements smaller or equal to the corresponding elements in the original error matrix, the difference representing the effect of knowing or not knowing the true value of the parameter that was removed at step two. This procedure is exactly that performed by Minuit when a FIX command is executed. Note that it is not reversible, since information has been lost in the deletion. The Minuit commands [REStore]RESTORE and [RELease]RELEASE therefore cause the error matrix to be considered lost and it must be recalculated entirely.