Showing a tensor is symmetric, nondegenerate and finding its inverse

[quasar_4]

Homework Statement

Let {e1, e2, e3} be a basis for vector space V. Show that the rank 2 tensor g defined by g=2E1*E2 + 2E2*E1+E1*E3+E3*E1 (where Ei are dual vectors and * is the tensor product) is symmetric and nondegenerate. Caculate g inverse.

Homework Equations

Um. lots of tensor stuff. I have the properties of a tensor, that's about it.

The Attempt at a Solution

I am wondering how I would show that it is non-degenerate. I know that if a bilinear form B is non-degenerate then: ker[Bv1]={x|B(x,y)=0 for all y} and ker[Bv2]={y|B(x,y)=0 for all x}.So we have to show that the kernels of the two vector spaces generating the bilinear form are 0 for all y and all x, respectively. But how on Earth is this done, especially when all I know is that we are taking two vectors from the vector space V for our tensor?

I don't get it! I CAN show that it is symmetric, I think, by evaluating g on the basis vectors one at a time with tensor properties (is that the right thing to do?) So to show symmetric I just took g(e1), g(e2), g(e3).

I haven't the foggiest how to find the inverse components.

 

[Avodyne]

It seems to me that your problem is equivalent to showing that the matrix

0 2 0
2 0 1
0 1 0

is symmetric, and does not have a zero eigenvalue.

 

[quasar_4]

aha, I have figured it out. There was a typo in the problem, which I resolved with the instructor. THe problem was that it was symmetric, but not non-degenerate (the determinant is zero. This is a problem to show non-degeneracy!) Once the problem is fixed, I guess the answer is to write out the matrix for it, show symmetric, show invertibility (this should show us non-degeneracy?), get the inverse components from that.

Thanks!