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where <math>\mathbf J</math> is the [[Jacobian matrix and determinant|Jacobian matrix]], whose {{tmath|i}}-th row equals <math>\mathbf J_i</math>, and where <math>\mathbf f\left (\boldsymbol\beta\right )</math> and <math>\mathbf y</math> are vectors with {{tmath|i}}-th component
<math>f\left (x_i, \boldsymbol\beta\right )</math> and <math>y_i</math> respectively. It should be noted that the Jacobian matrix as defined above is not (in general) a square matrix, but a rectangular matrix of size <math>m \times n</math>, where <math>n</math> is the number of parameters (size of the vector <math>\boldsymbol\beta</math>). The matrix multiplication <math>\left (\mathbf J^{\mathrm T} \mathbf J\right)</math> yields the required <math>n \times n</math> square matrix and the matrix-vector product on the right hand side yields a vector of size <math>n</math>.
Levenberg's contribution is to replace this equation by a "damped version":
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