Understanding Hessian for multidimensional function

  • #1
SaschaSIGI
3
0
Hello everybody,

I have a question regarding this visualization of a multidimensional function. Given f(u, v) = e^{−cu} sin(u) sin(v). Im confused why the maximas/minimas have half positive Trace and half negative Trace. I thought because its maxima it only has to be negative. 3D vis

2D visualization
 

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  • #2
Hi,

You have me wondering what I am looking at. Is the Hessian projected as a color code on a plot of the function ?
Did it occur to you to write down the Hessian for this function ? So: what's the expression for the trace of the Hessian ? (*)

What do you mean with
SaschaSIGI said:
because its maxima it only has to be negative

Aren't there minima between the maxima ?

(by the way: single: minimum, maximum. Plural: minima, maxima)

(*) Notice the similarity with the Laplacian :smile: ?

##\ ##
 

1. What is the Hessian matrix?

The Hessian matrix is a square matrix that contains the second-order partial derivatives of a multivariable function. It is used to determine the curvature and extrema of a function.

2. How is the Hessian matrix calculated?

The Hessian matrix is calculated by taking the second-order partial derivatives of a function and arranging them in a matrix form. The diagonal elements of the matrix represent the second-order partial derivatives with respect to each variable, while the off-diagonal elements represent the mixed partial derivatives.

3. What is the significance of the Hessian matrix in optimization?

The Hessian matrix is used in optimization to determine the nature of the critical points of a function. The positive definiteness of the Hessian matrix at a critical point indicates a local minimum, while a negative definiteness indicates a local maximum. A Hessian matrix with both positive and negative eigenvalues indicates a saddle point.

4. How does the Hessian matrix help in understanding multidimensional functions?

The Hessian matrix provides valuable information about the curvature and extrema of a multidimensional function. By analyzing the eigenvalues and eigenvectors of the Hessian matrix, we can determine the nature of critical points and understand the behavior of the function in different directions.

5. Can the Hessian matrix be used for non-differentiable functions?

No, the Hessian matrix can only be used for differentiable functions. It requires the existence of second-order partial derivatives, which do not exist for non-differentiable functions.

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