Second, you got the determinant of the Hessian matrix to be $$40x^{2}y^{-10}-64x^{2}y^{-10}=-24x^{2}y^{-10}\leq 0$$ and you concluded that the function was "concave". While the expression you had for the determinant of the Hessian is correct, your conclusion needs re-considerations.

2020年5月13日 · The Hessian. Simply, the Hessian is the matrix of second order mixed partials of a scalar field. $$\mathbf{H}_{i, j}=\frac{\partial^{2} f}{\partial x_{i} \partial x_{j}}$$ In summation: Gradient: Vector of first order derivatives of a scalar field. Jacobian: Matrix of gradients for components of a vector field

This is an old discussion and I would not intend to correct anything. Just keep in mind this had caused myself long-time confusion so I wish to accept some simple rules to make it clear.

Because the Hessian matrix is real and symmetric, we can decompose it into a set of real eigenvalues and an orthogonal basis of eigenvectors. The second derivative in a specific direction

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The Hessian matrix describes the curvature just when the gradient of f vanishes ("Relation to principal ...

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2018年11月20日 · Then, I would find the second order partial derivatives of f, and place them accordingly into a 2x2 Hessian Matrix A (i.e. f_xx as a_11, f_xy as a_12, f_yx as a_21, fyy as a_22). From here, I would sub in the critical point (granted …

If you want to denote an entry of the Hessian matrix (this is a second partial derivative of a function ...