# Questions tagged [hessian-matrix]

The Hessian matrix of function is used to second derivative test when $f$ has a critical point $x$. If the Hessian is positive definite at $x$, then $f$ attains a local minimum at $x$. If the Hessian is negative definite at $x$, then $f$ attains a local maximum at $x$. If the Hessian has both positive and negative eigenvalues then $x$ is a saddle point for $f$.

865 questions
Filter by
Sorted by
Tagged with
325 views

• 3,492
26 views

### What is the intepretation of a Hessian with only negative and zero eigenvalues?

I'm looking at a high-dimensional optimisation problem. Specifically, one involving a function $F(c_n)$ which depends on N parameters (in my case $N=200$). I have been able to minimise this function ...
26 views

• 31
73 views

### Derivatives of multivariate Gaussian

I do not understand how to take the second derivative of the Gaussian: While I am certain that the result is correct, I do not understand how to get from the first to the second line of the second-...
• 1,131
47 views

### Jacobian and Hessian of a vector

I am dealing with vector derivatives, and I am having a hard time generalising results. What I mean is that I can compute the results by hand, variable per variable, but I would like to do it in a ...
83 views

• 1,231
21 views

27 views

• 2,674
1 vote
50 views

### Relation between Hessian of the squared distance function and the metric tensor

Let $(M, g)$ be a Riemannian manifold and let $d$ be the distance function associated with the metric tensor $g$. Define the function $$f(x) := \frac12 d^2 (x,y)$$ for some arbitrary fixed $y \in M$....
• 37
47 views

### Classifying critical points when hessian has det0 [closed]

I am trying to find the critical points of $$f(x,y)=(x^2-y^4)(1-x^2-y^4).$$ There are 9, and I found all their coordinates. I’ve classified 6/9 using the hessian, but the remaining 3 have at least one ...
71 views

### Possible to have only zero eigenvalues of the Hessian of a harmonic function that is neither of the form $ax+by+cz+d$ nor a constant?

(Following an earlier post here) I intuit that if we restrict to functions $f(x,y,z)$ that are harmonic (i.e. satisfying $\nabla^2f=0)$ but neither of the form $ax+by+cz+d$ ($a,b,c,d\in\mathbb{R}$) ...
77 views

### Proving nonexistence of local extrema of harmonic functions using Hessian

Edited I want to prove that solution to Laplace's equation do not support local maximum or minimum using Hessian. Suppose $f(x,y,z)$ is a real-valued function of three real variables that satisfies ...
41 views

### Is this function convex over a set?

Well, I got a doubt: is this function $$f(x, y, z) = 2xy + yz$$ Convex over the set $$V =\{ (x,y,z) \in \mathbb\{R\}^3:\ x+y+2z \leq 1, x\geq 0, y \geq 0, z \geq 0\}$$ ? Being it a quadratic form, I ...
• 3,492
1 vote
89 views

• 65
1 vote
36 views

### Understanding derivation of Hessian Sketch formula

I've been reading this paper and in equation 23 they ask us to consider the optimization problem. \begin{align} \hat{u} = \arg \min_{u \in C - x^t} \left\{ \frac{1}{2} \lVert Au \rVert^2_2 - \langle A^...
1 vote
68 views

### What is the name of the middle matrix in a quadratic form $x'Ax$?

In a quadratic form, $x'Ax$, does matrix $A$ have a special name? I am trying to write a sentence that looks like "Using vector $x$ and matrix $A$ as the ______ matrix, we can write the quadratic ...
• 21
207 views

### Monotonicity of a multivariate function

Let $f:\mathbb R^n\times \mathbb R^n\rightarrow \mathbb R$. Assume $f(x,y)$ is twise continuously differentiable and the derivative with respect to both arguments (i.e., $x$ and $y$) are monotone in ...
50 views

• 25
44 views

• 1,419