The minimum norm Total Least Squares (TLS) solution to the linear system of equations AX = B for the case where the rank of the composite matrix [A I B] under no-noise/error-free conditions is strictly less than the number of columns in A is derived. The resulting method of solution is applied to the covariance level ESPRIT problem encountered in the field of sensor array signal processing. The TLS concept is targeted in light of the fact that the linear system of equations derived from ESPRIT at the covariance level is of the form AX = B where A and B are both in error. In contrast, conventional Least Squares only accounts for errors in B. The derivation is based on a projection operator interpretation of TLS1 analogous to the situation with conventional Least Squares in which the solution to AX = B is obtained by first projecting the columns of B onto range CA). The projection operator approach is further utilized to derive the constrained TLS solution when certain columns of A are known to be exact, i. e., free of error. The applicability of the constrained TLS solution for this case to the problem of estimating the source covariance matrix in a uniformly-spaced linear array scenario is discussed.