Hessian as "Square" of Jacobian?

In summary, the conversation discusses the representation of the Laplacian for two variables and whether it can be expressed as a "square of Jacobians". The speaker also mentions using a rotation matrix to show the Laplacian's rotational invariance. The other person corrects them on the notation for the Hessian and provides the correct formula for it.
  • #1
WWGD
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Hi,
Is there a way of representing the Laplacian ( Say for 2 variables, to start simple) ##\partial^2(f):= f_{xx}+f_{yy} ## as a "square of Jacobians" ( More precisely, as ##JJ^T ; J^T ## is the transpose of J, for dimension reasons)? I am ultimately trying to use this to show that the Laplacian is rotationally-invariant, using a rotation matrix and manipulating the product.
 
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  • #2
Don't understand. ##\Delta \equiv \nabla^2## but even for 2 variables the Hessian is not as you write it !? it is the matrix product of ##\nabla## and ##\nabla##: ##H_{ij} = \nabla_i\nabla_j##

Ah, my bad: Hessian = det##H##
 
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Related to Hessian as "Square" of Jacobian?

1. What is the Hessian matrix and how is it related to the Jacobian matrix?

The Hessian matrix is a square matrix of second-order partial derivatives of a multivariable function. It is closely related to the Jacobian matrix, which is a matrix of first-order partial derivatives. The Hessian matrix can be thought of as the square of the Jacobian matrix.

2. How is the Hessian matrix used in optimization problems?

The Hessian matrix is used in optimization problems to determine the curvature of a function at a specific point. By analyzing the eigenvalues of the Hessian matrix, we can determine whether a critical point is a maximum, minimum, or saddle point. This information is crucial in finding the optimal solutions to optimization problems.

3. Can the Hessian matrix be used for functions with multiple variables?

Yes, the Hessian matrix can be used for functions with multiple variables. It is a square matrix with the same number of rows and columns as the number of variables in the function. The Hessian matrix is used to analyze the curvature of a multivariable function and determine the critical points.

4. How is the Hessian matrix calculated?

The Hessian matrix is calculated by taking the second-order partial derivatives of a function with respect to each variable and arranging them in a specific way. For a function with n variables, the Hessian matrix will be an n x n matrix. The elements of the Hessian matrix can then be used to analyze the curvature of the function and find critical points.

5. Are there any limitations to using the Hessian matrix as the square of the Jacobian matrix?

While the Hessian matrix is a useful tool in optimization problems, it is not always applicable. It can only be used for functions with continuous second-order partial derivatives. Additionally, the Hessian matrix may not be defined at every point, so it is important to check for its existence before using it in calculations.

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