What is Hessian: Definition and 50 Discussions

Hessians (US: or UK: ) were German soldiers who served as auxiliaries to the British Army during the American Revolutionary War. Britain had a small army, most of which were needed at home. It decided to rent regiments. Most governments refused, but several small German states such as Hesse had a reputation as "mercenary states" and rented regiments to the British for combat duty. The Patriots often called the hired German soldiers "Hessians" and denounced them all as "mercenaries" who were not fighting for their native land.
In his history of the Hessians, Rodney Atwood says, "By common usage, however, the Hessians have been called mercenaries. In this work I refer to them as both auxiliaries and mercenaries."
The term is an American synecdoche for all Germans who fought on the British side, since 65% came from the German states of Hesse-Kassel and Hesse-Hanau. Known for their discipline and martial prowess, around 30,000 Germans fought for the British during war, comprising a quarter of British land forces.Although frequently referred to by scholars as mercenaries, Hessians were in terms of 18th century legal thought distinguished as auxiliaries. Unlike mercenaries, who served a foreign government on their own accord, auxiliaries were soldiers hired out to a foreign party by their own government, to which they remained in service. As a source of funding throughout the 18th century, many small poor German states regularly rented out the services of their troops to fight in wars in which they were neutral and had no other involvement. Like most auxiliaries of this period, Hessians served with foreign armies as entire units, fighting under their own flags, commanded by their usual officers, and wearing their existing uniforms.
Hessians played a key role in the Revolutionary War. They served with distinction in many battles, particularly in the northern theater, most notably at White Plains and Fort Washington. The added manpower and skill of German troops is credited for greatly sustaining the British war effort, but it also outraged colonists and increased support for the Patriot cause. The use of "large armies of foreign mercenaries" was one of the 27 colonial grievances against King George III in the United States Declaration of Independence, while the Patriots
used the deployment of Hessians to support their claims of British violations of the colonist's rights.

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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. ergospherical

    I What is the Coordinate-Free Formulation of the Hessian?

    In local coordinates, the hessian of the function ##f## at point ##p## is ##H = \partial_i \partial_k f dx^i \otimes dx^k##. A coordinate-free generalisation is (see) ##H = \nabla df##, or explicitly ##H = \nabla_i (df)_k dx^i \otimes dx^k = \nabla_i \partial_k f dx^i \otimes dx^k##. How is...
  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. F

    I Relation with Hessian and Log-likelihood

    I would like to demonstrate the equation (1) below in the general form of the Log-likelihood : ##E\Big[\frac{\partial \mathcal{L}}{\partial \theta} \frac{\partial \mathcal{L}^{\prime}}{\partial \theta}\Big]=E\Big[\frac{-\partial^{2} \mathcal{L}}{\partial \theta \partial...
  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. W

    A Hessian as "Square" of Jacobian?

    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...
  7. aphirst

    A Derivative of argmin/argmax w.r.t. auxiliary parameter?

    As part of my work, I'm making use of the familiar properties of function minima/maxima in a way which I can't seem to find in the literature. I was hoping that by describing it here, someone else might recognise it and be able to point me to a citation. I think it's highly unlikely that I'm the...
  8. N

    SIFT is derivative of DoG needed for Hessian or just DoG?

    Wikipedia defines hessian of Difference of Gaussians as and earlier in the page uses D for difference of gaussians, So do i just need D(x,y) or do i need d/dx D(x,y) for the elements? If so how does one go about differentiating DoG? Any help appreciated
  9. Jeffack

    I Hessian of least squares estimate behaving strangely

    I am doing a nonlinear least squares estimation on a function of 14 variables (meaning that, to estimate ##y=f(x)##, I minimize ##\Sigma_i(y_i-(\hat x_i))^2## ). I do this using the quasi-Newton algorithm in MATLAB. This also gives the Hessian (matrix of second derivatives) at the minimizing...
  10. perplexabot

    A Calculating Hessian of f(x)^TQy: What Can We Conclude?

    Hey all. Let me just get right to it! Assume you have a function f:\mathbb{R}^n\rightarrow\mathbb{R}^m and we know nothing else except the following equation: \triangledown_x\triangledown_x^Tf(x)^TQy=0 where \triangledown_x is the gradient with respect to vector x (outer product of two gradient...
  11. D

    Gradient and Hessian of the Coulomb/Electrostatic Energy

    I have a function $$\displaystyle V(x)=\frac{1}{2}\sum_i \sum_{j \neq i} q_i q_j \frac{1}{\left|r_i - r_j\right|}$$ where ##r_i=\sqrt{x_i^2+y_i^2+z_i^2}## which is the coulomb potential energy of a system of charges. I need to calculate ##\frac{\partial V}{\partial x_k}## and...
  12. P

    Optimization using Newton's method gradient hessian

    Hello, This is my first post here. So I hope I'm posting in the right place, sorry if not. http://homes.soic.indiana.edu/classes/spring2012/csci/b553-hauserk/Newtons_method.pdf I am trying to solve the following numerical optimization function using Netwon's Method: So, if I have the gradient...
  13. 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) =...
  14. 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...
  15. 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...
  16. quasar987

    What Conditions Make a Matrix the Hessian of a Function?

    Apparently it is a well-known fact that if G(x)=(G_{ij}(x_1,\ldots,x_n)) is a smooth nxn matrix-valued function such that G_{ij,k}=G_{ik,j} for all i,j,k, then there exists a smooth function g such that Hess(g)=G; i.e. g_{,ij}=G_{ij}. (f_{,k} denotes partial differentiation with respect to the...
  17. 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...
  18. P

    Basic question about numerical hessian

    Hello all, Suppose I have a simple 1-D signal and I want to compute the hessian. In that case, it should generalise for second derivative for normal scalar functions. So, I observe the signal as v = [x_1, x_2, x_3, x_4...]. Then, numerically the hessian is given as (assuming I am only...
  19. J

    Understanding Matrix Calculus: Laplacian, Hessian, and Jacobian Explained

    Hellow! I was studying matrix calculus and learned new things as: \frac{d\vec{y}}{d\vec{x}}=\begin{bmatrix} \frac{dy_1}{dx_1} & \frac{dy_1}{dx_2} \\ \frac{dy_2}{dx_1} & \frac{dy_2}{dx_2} \\ \end{bmatrix} \frac{d}{d\vec{r}}\frac{d}{d\vec{r}} = \frac{d^2}{d\vec{r}^2} = \begin{bmatrix}...
  20. P

    Visualizing second derivative test - Hessian

    Hey guys. I am having some trouble visualizing one aspect of the Second derivative test in the 2 variable case (related to #3 below). Essentially, what does the curve look like when f_{xx}f_{yy} > 0, BUT f_{xx}f_{yy} < [f_{xy}]^{2}? To be more detailed, if the function is f(x,y), H(x,y) is the...
  21. M

    Finding the Nature of Critical Points Using Hessian

    Homework Statement Identify and determine the nature of the critical points of the function $$f(x,y,z) = (x^2 + 2y^2 + 1) cos z$$ Homework Equations ##\vec{x}## is a critical point ##\iff Df(\vec{x}) = 0## ##\vec{x}## is a minimum ##\iff## every determinant of upper left submatrix...
  22. L

    Convexity of a functional using the Hessian

    Homework Statement Consider the functional I:W^{1,2}(\Omega)\times W^{1,2}(\Omega)\rightarrow \mathbb{R} such that I(f_1,f_2)=\int_{\Omega}{\dfrac{1}{2}|\nabla f_1|^2+\dfrac{1}{2}|\nabla f_1|^2+e^{f_1+f_2}-f_1-f_2}dx. I would like to show that the functional is strictly convex by using the...
  23. 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...
  24. C

    Difference between a hessian and a bordered hessian

    Homework Statement I was wondering what exactly the difference between a regular (proper? is that the term) hessian is and a bordered hessian. It is difficult to find material in the book or online at this point. I mean mathmatically so that were i to do a problem i would know the layout and...
  25. O

    MHB What is the Hessian method for determining concavity/convexity?

    Hello Everyone! I'm trying to remember a quick method for determining whether a function is concave or convex. There was something that involved finding the Hessian of the function, and then looking at the diagonal elements, then, I completely forgot... What's the rest of this method, I don't...
  26. 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...
  27. K

    Leading Principal Minors of Bordered Hessian in Constrained Max Problems

    I am struggling a bit with the second order conditions of a constrained maximization problem with n variables and k constraints (with k>n). In the equality constraints case we have to check if the (n-k) leading principal minors of the bordered Hessian alternate in sign, starting from the...
  28. S

    Why does the Hessian determinant Δ_p = -1 imply that P(0, 0) is a saddle point?

    Homework Statement For the function f(x, y) = xye^[-(x^2 + y^2)] find all the critical points and classify them each as a relative maximum, a relative minimum, or a saddle point. Homework Equations Partial differentiation and Hessian determinants. The Attempt at a Solution I get how...
  29. 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...
  30. K

    Show Hessian is positive definite

    Homework Statement Consider the function f(a)= 1 ∫ [g(x)-(anxn+an-1xn-1+...+a0)]2 dx 0 where a=(a0,a1,...an) and g is some known function defined on [0,1]. From this, we can show that Thus, the Hessian of f at a = [2/(j+k+1)] j=0,1,2,...n; k=0,1,2,...,n. Fact: This Hessian...
  31. 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...
  32. B

    Optimization problem using exact Hessian

    hi, I'm kind of new to optimization theory, and I have to maximize a multi-dimensional problem where I know the exact gradient and hessian. In other words, techniques such as BFGS are not sufficient because I don't want to approximate the Hessian (with an initial guess for example of H=I), I...
  33. Y

    What Are the Best Resources for Understanding Jacobian and Hessian Matrices?

    Can someone direct me to a good deep exposition of Jacobians and Hessians? I am especially looking for stuff that pertains to their being generalizations of derivatives of vector and scalar functions as well as div, grad, curl. Book sources or web links are appreciated.
  34. 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...
  35. B

    Mathematica Hessian optimization through Mathematica

    I know by default that Mathematica will use the BFGS method when you request "FindMinimum[Function]" but I am curious for a hint towards a pseudo-code for the following problem: I have a collection of functions, say F = {f1,f2,...,fN} and I want to transform them as linear combinations of one...
  36. 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...
  37. E

    The Gradient and the Hessian of a Function of Two Vectors

    Hi, Suppose we have a function of two n-dimensional vectors f(\mathbf{x},\mathbf{y}). How can we find the gradient and Hessian of this function? Regards
  38. 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...
  39. L

    Coordinate free definition of hessian

    Let M be a manifold and f: M \rightarrow \mathbb{R} be a smooth function such that df=0 at some point p \in M. Let \{ x^\mu \} be a coordinate chart defined in a neighbourhood of p. Define F_{\mu \nu} = \frac{ \partial f}{ \partial x^\mu \partial x^\nu } By considering the transoformation...
  40. 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...
  41. G

    Critical Point Classification: Inconclusive Hessian

    What is the general approach to take when the Hessian is inconclusive when classifying critical points? ie the determinant = 0?
  42. L

    Second derivative test when Hessian is Positive Semi-Definite

    Can someone tell me what this actually is. So, in the case when the Hessian is positive (or negative) semidefinite, the second derivative test is inconclusive. However, I think I've read that even in the case where the Hessian is positive semidefinite at a stationary point x, we can still...
  43. K

    How do you determine the behavior of critical points when you have the Hessian?

    Hello, I have solved for the critical points using the gradient, and I have solved for the Hession, which yields a 2x2 matrix. I have plugged in my critical points into the gradient. Now, do I apply the same rules as in linear algebra where I find the determinant and trace to calculate...
  44. 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?
  45. 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 ?
  46. 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...
  47. K

    Local Extrema, Hessian & Eigenvalues

    1) f(x,y,z)=x3-3x-y3+9y+z2 Find and classify all critical points. I am confused about the following: The Hessian matrix is diagonal with diagonal entries 6x, -6y, 2. Now, the diagonal entries of a diagonal matrix are the eigenvalues of the matrix. (this has to be true, it is already...
  48. 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...
  49. G

    What is the solution to finding the top of a hill using gradients and hessian?

    Homework Statement The height of a certain hill (in feet) is given by h(x,y) = 10(2xy-3x^2-4y^2-18x+28y+12) where y is the distance (in miles) north, x the distance east of South Hadley. a) Where is the top of the hill located? b) How high is the hill? c) How steep is the slope (in...
  50. A

    Hessian Matrix\Max Min Analysis, Eigenvalues etc

    In my calc 3 class, we've taken an alternative(?) route to learning maxes and mins of multivariable equations. By using a Hessian Matrix, we're supposed to be able to find the eigenvalues of a function at the point, and determine whether the point is a max, min, saddle point, or indeterminant...