Canonical form of a quadratic response function

The conversion in a chemical reaction was described by the empirical relationship (ref. 13) 

As we will see later, A is nonsingular, hence the only solution is 

What can we see from these results The point x° is not a maximum, since the first eigenvalue is positive. Selecting the canonical variables w 0, w2 = w3 = can increase the value of . By orthogonality of the 

To find the point xD one can use LU decomposition and a backsubstitution. An other possibility is to apply the results of the eigenvalue-eigenvector decomposition directly. By eqn. (1.66) 

Evaluating this expression is quite easy if taking into account that D is a diagonal matrix with reciprocal values of the eigenvalues in its main diagonal. 

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