A. N. Tikhonov’s lemma is related to the optimal matrix correction of an inconsistent (contradictory) system of linear algebraic equations. This paper presents an elementary proof of Tikhonov’s lemma relying solely on the Cauchy–Schwarz inequality for vectors and the condition under which the equality holds in that inequality. It is important that the formula for the optimal matrix is not simply “proposed and verified,” but is actually derived. This work considers V.I. Erokhin’s generalization of Tikhonov’s lemma to the case of a matrix correction of an inconsistent system consisting of two conjugate systems of linear equations. Explicit formulas are derived for the optimal matrix and the square of its norm. A factorization of the optimal matrix is obtained. Using this factorization, the rank of the optimal matrix is calculated. The optimal matrix is expressed as the difference of two matrices: that for which both conjugate systems of linear equations are consistent, and the matrix orthogonal to the optimal matrix.
Malozemov et al. (Sun,) studied this question.