\left | H_n \right \vert = \begin{vmatrix}  \frac{\partial^2 f}{\partial {x_1}^2}(\mathbf{x^*}) & \frac{\partial^2 f}{\partial {x_1}\,\partial {x_2}}(\mathbf{x^*}) & \cdots & \frac{\partial^2 f}{\partial {x_1}\,\partial x_n}(\mathbf{x^*}) \  \  \frac{\partial^2 f}{\partial {x_2}\,\partial {x_1}}(\mathbf{x^*}) & \frac{\partial^2 f}{\partial {x_2}^2}(\mathbf{x^*}) & \cdots & \frac{\partial^2 f}{\partial {x_2}\,\partial x_n}(\mathbf{x^*}) \  \  \vdots & \vdots & \vdots & \vdots \ \  \frac{\partial^2 f}{\partial x_n\,\partial {x_1}}(\mathbf{x^*}) & \frac{\partial^2 f}{\partial x_n\,\partial {x_2}}(\mathbf{x^*}) & \cdots & \frac{\partial^2 f}{\partial x_n^2}(\mathbf{x^*}) \end{vmatrix} = \left | H_n \right \vert
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