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The existence of a nonsingular matrix is proved for any space of square symmetric matrices with a trivial quadratic kernel. Some corollaries from this result are obtained for construction of solvers of nonlinear equations and problems of conditional optimization with a Jacobi matrix of incomplete rank based on the theory of  p -regularity for  p  = 2. Some remarks for the case  p  ⩾ 3 are presented.

The triviality condition for kernels of quadratic mappings and its application in optimization methods