What is Hessian matrix: Definition and 23 Discussions

In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named after him. Hesse originally used the term "functional determinants".

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  1. SaschaSIGI

    I Understanding Hessian for multidimensional function

    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...
  2. F

    Stationary points classification using definiteness of the Lagrangian

    Hello, I am using the Lagrange multipliers method to find the extremums of ##f(x,y)## subjected to the constraint ##g(x,y)##, an ellipse. So far, I have successfully identified several triplets ##(x^∗,y^∗,λ^∗)## such that each triplet is a stationary point for the Lagrangian: ##\nabla...
  3. M

    MHB F convex iff Hessian matrix positive semidefinite

    Hey! A function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is convex if for all $x,y\in \mathbb{R}^n$ the inequality $$f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)$$ holds for all $t\in [0,1]$. Show that a twice continuously differentiable funtion $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is convex iff the...
  4. A

    A Correlation between parameters in a likelihood fit

    Hello community! I am facing a conceptual problem with the correlation matrix between maximum likelihood estimators. I estimate two parameters (their names are SigmaBin0 and qqzz_norm_0) from a multidimensional likelihood function, actually the number of parameters are larger than the two I am...
  5. M

    MHB Unfamiliar hessian matrix expression

    I am familiar with the hessian matrix having the square in the numerator and a product of partial derivatives in the denominator: $Hessian = \frac{\partial^2 f}{\partial x_i \partial x_j}$ However, I have come across a different expression, source...
  6. KareemErgawy

    Calculating mixed partial derivatives on a 3D mesh

    I am working on implementing a PDE model that simulates a certain physical phenomenon on the surface of a 3D mesh. The model involves calculating mixed partial derivatives of a scalar function defined on the vertices of the mesh. What I tried so far (which is not giving good results), is this...
  7. Coffee_

    Proof: extremum has a semi definitie Hessian matrix

    Consider a function ##f : U \subseteq \mathbb{R}^{n} -> \mathbb{R}## that is an element of ##C^{2}## which has an minimum in ##p \in U##. According to Taylor's theorem for multiple variable functions, for each ##h \in U## there exists a ##t \in ]0,1[## such that : ##f(p+h)-f(p) =...
  8. kostoglotov

    Q about 2nd derivative test for multivariable functions

    Homework Statement So the test is to take the determinant (D) of the Hessian matrix of your multivar function. Then if D>0 & fxx>0 it's a min point, if D>0 & fxx<0 it's a max point. For D<0 it's a saddle point, and D=0 gives no information. My question is, what happens if fxx=0? Is that...
  9. W

    Is the Hessian Matrix anything more than a mnemonic?

    Several questions I have been thinking about... let me know if you have thoughts on any of them I added numbers to for coherence and readability. So, the Hessian matrix can be used to determine the stability of critical points of functions that act on \mathbb{R}^{n}, by examining its...
  10. C

    Hessian matrix of the Newtonian potential is zero?

    So I'm looking at the hessian of the Newtonian potential: \partial^2\phi / \partial x_i \partial x_j Using the fact that (assuming the mass is constant): F = m \cdot d^2 x / d t^2 = - \nabla \phi This implies: \partial^2\phi / \partial x_i \partial x_j = -m \cdot...
  11. C

    Second derivative test and hessian matrix

    How does one derive the second derivative test for three variables? It's clear that D(a,b) = fxx * fyy - (fxy)^2 AND fxx(a,b) Tells us almost all we need to know about local maxima and local minima for a function of 2 variables x and y, but how do I make sense of the second directional...
  12. F

    Hessian matrix of potential energy in electrostatic system

    Hi everyone: I am rookie in classical physics and first-time PF user so please forgive me if I am making mistakes here. My current project needs some guidance from physics and I am describing the problem, my understanding and question as below. I have an independent electrostatic system...
  13. S

    Optimization & singular Hessian matrix

    I am trying to figure out how the least squares formula is derived. With the error function as Ei = yi - Ʃj xij aj the sum of the errors is SSE = Ʃi Ei2 so the 1st partial derivative of SSE with respect to aj is ∂SSE / ∂aj = Ʃi 2 Ei ( ∂Ei / ∂aj ) with the 1st partial derivative of...
  14. S

    Hessian matrix in taylor expansion help

    Homework Statement Find the critical point(s) of this function and determine if the function has a maxi- mum/minimum/neither at the critical point(s) (semi colons start a new row in the matrix) f(x,y,z) = 1/2 [ x y z ] [3 1 0; 1 4 -1; 0 -1 2] [x;y;z] Homework Equations The...
  15. A

    Why Must the Hessian Matrix Be Symmetric at a Critical Point?

    Homework Statement Given a function f: R^2 -> R of class C^3 with a critical point c. Why CANNOT the hessian matrix of f at point c be given by: 1 -2 2 3 Homework Equations The Attempt at a Solution So first i want to clarify this. When it says f: R^2 -> R, that...
  16. T

    Finding multivariate extrema with degenerate hessian matrix

    Homework Statement For what real values of the parameters a,b,c,d does the functiob f(x,y)=ax^3+by^3+cx^4+dy^4-(x+y)^5 have a local minimum at (0,0)Homework Equations I calculated the gradient at (0,0) and it is always zero regardless of parameters. The problem is that the Hessian matrix is...
  17. D

    Proving extrema using taylor series and Hessian Matrix

    How do I use Taylor Series to show f(P) is a local maximum at a stationary point P if the Hessian matrix is negative definite. I understand that some of the coefficients of the terms of the taylor series expansion are the coordinates of the Hessian matrix but for the f_xy term there is no...
  18. J

    X Vector in 2nd Order Taylor Series Formula w/ Hessian Matrix

    The formula given by my instructor for a Taylor Series approximation of the second order at point (a,b) is f(a,b) + grad(f(a,b))x + 1/2 H(f(a,b)) x If you recognize this formula, do you know what the x vector is? Note: x is the x-vector, and H represents the Hessian Matrix. Thanks! The...
  19. Telemachus

    Hessian matrix, maxima and minima criteria

    Homework Statement Hi there. I've got some doubts about the maxima and minima on this function: f(x,y)=x \sin y. I've looked for critical points, and there's only one at (0,0). The thing is that when I've evaluate the second derivatives I've found that f_{xx}=0, then I have not a defined...
  20. P

    Function two wariables - hessian matrix is 0

    Homework Statement what can I do if I have hessian = 0? ex. function f(x,y)=x^2+y^4 hessian is 0, what now? this is simply but what can i do in more complicated functions?
  21. A

    Exploring the Uses of Hessian Matrix in Multivariable Calculus

    What's Hessian matrix ? Here are all my problem ~ 1. What's Hessian matrix ? 2. How Hessian matrix was derived ? 3. Can u recommend some books about this ?
  22. F

    Solving Eigenvalues of Hessian Matrix

    g(x,y) = x^3 - 3x^2 + 5xy -7y^2 Hessian Matrix = 6x-6******5 5********-7 Now I have to find the eigenvalues of this matrix, so I end up with the equation (where a = lambda) (6x - 6 - a)(-7 - a) - 25 = 0 Multiplying out I get: a^2 - 6xa + 13a - 42x + 17 = 0 How am I supposed to solve...
  23. A

    Find Critical Points of Hessian Matrix

    Please,check my solution. Find critical points of the function f(x,y,z)=x^3+y^2+z^2+12xy+2z and determine their types (degenerate or non-degenerate, Morse index for non- degenerate). Attempt \frac{df}{dx}=3x^2+12y=0 \frac{df}{dy}=2y+12x=0 \frac{df}{dz}=2z+2=0 Critical points...