The question reads:
Let be vectors in , and assume that . Prove that there is a symmetric matrix such that .
I don't really know where to begin on this. It's in the chapter on bilinear forms, but I don't see what this has to do with that. Maybe find some vector such that , so that we have to solve , in other words, show that there exists a vector that is orthogonal to with respect to the bilinear form (and then maybe use the fact that the eigenvectors of form an orthogonal basis).
But other than that speculation, I got nothing.
Help is appreciated!
I agree. That is simple. It's strange that they put this in a section on bilinear forms and marked it as a challenge problem.
Nonetheless, I'm more than happy to see that this has a simple solution.
Thanks for the help!
If anyone knows how to do this using bilinear forms, I'd love to know.
Let be the complex matrix space, be the complex vector space.
Define for any B in the matrix space, x in the vector space. It is obviously a biliear function.
For any fixed B, it defines a liear transform in the complex vector space;
For any fixed x, it defines a action (linear transform) on the complex matrix space, Since x is not 0, there is a base which contains x, Obviously there is a symmetric matrix B such that Bx=y, if we consider B as the matrix of a symmetric linear transform with respect to the given base.