What Conditions Make a Matrix the Hessian of a Function?

[quasar987]

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 kth variable.)

I believe I can construct the solution explicitly in the n=2 case, but I'm not sure how to generalize my argument. Is there an argument to be made about the existence of a solution to this overdetermined system of PDE? Thx!

 

[jvicens]

Thank you for bringing up this interesting topic. The statement you mentioned is indeed a well-known fact in mathematics and is known as the "Poincaré lemma". It states that if a smooth differential form satisfies certain conditions, then it can be locally written as the exterior derivative of another form.

In your case, the matrix-valued function G satisfies a certain compatibility condition, which is equivalent to saying that it can be written as the Hessian of a smooth function g. This is a special case of the Poincaré lemma, where the differential form is the Hessian and the compatibility condition is known as the integrability condition.

To answer your question about generalizing this argument, the Poincaré lemma can be extended to higher dimensions and more general settings. For example, in the n=3 case, the Hessian matrix G will be a 3x3 matrix and the corresponding integrability condition will involve the third derivatives of g. In higher dimensions, the integrability condition will involve even higher derivatives of g. However, the basic idea remains the same - if the compatibility condition is satisfied, then there exists a smooth function g whose Hessian is G.

As for the existence of a solution to this overdetermined system of PDE, it is important to note that the Poincaré lemma guarantees the existence of a local solution. In other words, there exists a smooth function g in a small neighborhood around any given point. However, the existence of a global solution is not guaranteed and depends on the specific form of G. In some cases, it may be possible to find a global solution, while in others it may not be possible.

I hope this helps to clarify your doubts. If you have any further questions, please don't hesitate to ask. it is always important to question and seek deeper understanding of concepts. Keep up the good work!