Jacobi Matrix and Linearisation

In the analysis of nonlinear problems, the first step usually consists in that the problem is linearized. That means that nonlinear functions occurring in the problem are approximated by Taylor expansion up to the first degree. For a vector-valued function of several variables, the coefficients (partial derivatives) are elements of the Jacobi matrix. 

Let g be an A/-vector of scalar functions of N scalar variables, thus of iV-vector z. Then the Jacobi matrix of g, say G, is the M x iV matrix of elements 

Let Zq G For an arbitrary z g /, the Taylor formula (restricted to the first degree) reads 

In this manner, for example the equation (8.3.53) becomes a linear equation (7.1.1) and can be analyzed by the methods described in Chapter 7. 

Generally, the same set of constraints (representing the same feasible set 9d) can be formulated by an infinity of equivalent equations. While in linear algebra, the equivalence means simply a regular transformation (multiplying by a regular square matrix), this is not the case when nonlinearity is admitted. Then not any (though equivalent) formulation of the model is equally appropriate for the solvability analysis. Observe that in (7.1.4), we assumed that the matrix C was of full row rank. It is thus natural to require also in the present case that 

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