The General Linear Model — a Solution by Means of the Condition Adjustment

It is well known that the least squares estimation is equivalent to the Best Linear Unbiased Estimation (BLUE) in the case. that the covariance matrix of the observations is positive definite. If the variance covariance matrix is positive semi-definite the inverse of this matrix (the weight matrix) is in general not unique so that the question arises: which weight matrix has to be used in least squares estimation to get the BLUE? This question is discussed in this paper and an alternative solution by means of the condition adjustment is given. The numerical problems that can occur when the modified covariance matrix is used can be avoided by using the alternative solution. With respect to the theory presented in this paper, it is shown how important a correct definition of weight matrix in the least squares method is.