When there exist linear constraints among parameters in the linear statistical model, the method of least squares seeks for the values of parameters which minimize the sum of squares of errors subject to the given linear constraints. For this minimization the Lagrange multiplier method is commonly used. This paper treats of the problem of finding whether any Lagrange multipliers in the normal equations are equal to zero or not, before solving the normal equations.
A necessary and sufficient condition that the Lagrange multipliers corresponding to any subset of linear constraints can be taken to be equal to zero is given. A sufficient condition for any Lagrange multipliers to be equal to zero is also given. Further a method for finding zero Lagrange multipliers is proposed and is illustrated by some examples.