Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

When is a matrix a hessian?

  1. Mar 20, 2014 #1

    quasar987

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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!
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: When is a matrix a hessian?
  1. Hermetic matrix (Replies: 2)

Loading...