Orthogonal projection and total least squares

Overdetermined linear systems often arise in applications such as signal processing and modern communication. When the overdetermined system of linear equations AX ≈︁ B has no solution, compatibility may be restored by an orthogonal projection method. The idea is to determine an orthogonal projection matrix P by some method M such that [ à B̃ ] = P [ A B ], and ÃX = B̃ is compatible. Denote by X M the minimum norm solution to ÃX = B̃ using method M. In this paper conditions for compatibility of the lower rank approximation and subspace properties of à in relation to the nearest rank‐ k matrix to A are discussed. We find upper and lower bounds for the difference between the solution X M and the SVD‐based total least squares (TLS) solution X SVD and also provide a perturbation result for the ordinary TLS method. These results suggest a new algorithm for computing a total least squares solution based on a rank revealing QR factorization and subspace refinement. Numerical simulations are included to illustrate the conclusions.