For a multivariate measurement error model, the authors consider the elementwise weighted total least squares (TLS) estimator. This problem covers the whole class of problems in which the errors in each element are proportional to its size and is therefore an important extension of the class of TLS problems studied so far. Some results proved by \textit{P. P. Gallo} [Commun. Stat. Theory Methods 11, 973--983 (1982; Zbl 0515.62064)] and \textit{L. J. Gleser} [Ann. Stat. 9, 24--44 (1981; Zbl 0496.62049)] are extended to the multivariate model. The approach is different. The errors are allowed to be unequally sized, consequently, it is impossible to derive an explicit formula for the estimator. It is shown that the estimator is the solution of an optimization problem. By analyzing the corresponding objective function, mild conditions for weak and strong consistency of the estimator are given. An iteratively reweighted numerical procedure is proposed.