[HESse]HESSE simply calculates the full second-derivative matrix by finite differences and inverts it. It therefore calculates the error matrix at the point where it happens to be when it is called. If the error matrix is not positive-definite, diagnostics are printed, and an attempt is made to form a positive-definite approximation. The error matrix must be positive-definite at the solution (minimum) for any real physical problem. It may well not be positive away from the minimum, but most algorithms including the [MIGrad]MIGRAD algorithm require a positive-definite ``working matrix''.
The error matrix produced by [HESse]HESSE is used to calculate what Minuit prints as the parameter errors, which therefore contain the effects due to parameter correlations. The extent of the two-by-two correlations can be seen from the correlation coefficients printed by Minuit, and the global correlations (see [5], p. 23) are also printed. All of these correlation coefficients must be less than one in absolute value. If any of them are very close to one or minus one, this indicates an illposed problem with more free parameters than can be determined by the model and the data.