What Conditions Make a Matrix the Hessian of a Function?

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Thread starter quasar987
Start date Mar 20, 2014
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Hessian Matrix

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SUMMARY

The discussion centers on the conditions under which a smooth nxn matrix-valued function G(x) can be the Hessian of a function g. It is established that if G_{ij,k} = G_{ik,j} for all indices i, j, and k, then a smooth function g exists such that Hess(g) = G, specifically g_{,ij} = G_{ij}. The participant expresses confidence in constructing a solution for the case n=2 but seeks guidance on generalizing this to higher dimensions and addressing the overdetermined system of partial differential equations (PDEs).

PREREQUISITES

Understanding of smooth functions and their properties
Familiarity with Hessian matrices in multivariable calculus
Knowledge of partial differential equations (PDEs)
Experience with matrix calculus and differentiation

NEXT STEPS

Research the theory behind Hessian matrices in higher dimensions
Explore methods for solving overdetermined systems of PDEs
Study examples of constructing smooth functions from given Hessians
Learn about the implications of symmetry in matrix-valued functions

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Mathematicians, particularly those specializing in differential equations, multivariable calculus, and theoretical physics, will benefit from this discussion.

Mar 20, 2014

quasar987

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Apparently it is a well-known fact that if [itex]G(x)=(G_{ij}(x_1,\ldots,x_n))[/itex] is a smooth nxn matrix-valued function such that [itex]G_{ij,k}=G_{ik,j}[/itex] for all i,j,k, then there exists a smooth function g such that Hess(g)=G; i.e. [itex]g_{,ij}=G_{ij}[/itex]. ([itex]f_{,k}[/itex] 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!