Total least-squares adjustment of condition equations

The usual least-squares adjustment within an Errors-in-Variables (EIV) model is often described as Total Least-Squares Solution (TLSS), just as the usual least-squares adjustment within a Random Effects Model (REM) has become popular under the name of Least-Squares Collocation (without trend). In comparison to the standard Gauss-Markov Model (GMM), the EIV-Model is less informative whereas the REM is more informative. It is known under which conditions exactly the GMM or the REM can be equivalently replaced by a model of condition equations or, more generally, by a Gauss-Helmert Model. Similar equivalency conditions are, however, still unknown for the EIV-Model once it is transformed into such a model of condition equations. In a first step, it is shown in this contribution how the respective residual vector and residual matrix look like if the TLSS is applied to condition equations with a random coefficient matrix to describe the transformation of the random error vector. The results are demonstrated using a numeric example which shows that this approach may be valuable in its own right.