Tensor Mechanics Problem

[Joang]

Please, colleagues, I would greatly appreciate your assistance with the following problem.



Or any criteria that may help solve it
Thank you

 

[blamocur]

This is way too vague. What are [imath]a_{\alpha\beta}[/imath], [imath]x^\alpha[/imath] and [imath]y^\beta[/imath] ? For example, in [imath]x^\alpha[/imath] is [imath]\alpha[/imath] a superscript or is it [imath]x[/imath] to the [imath]\alpha[/imath]-th power ? Please provide a fuller description.
Thank you.

 

[blamocur]

Here is what I think your post means: you are given a bilinear form in Einstein notation. In my preferred notation it looks like this: [imath]S(\mathbf x, \mathbf y) = \mathbf x^T \mathbf A \mathbf y = \sum_{i,j} x_i a_{ij} y_j[/imath] and condition [imath]S(\mathbf x, \mathbf x) = \sum_{i,j} x_i a_{ij} x_j = 0[/imath]. Then you need to show that [imath]a_{ij} = - a_{ji}[/imath], i.e., that [imath]\mathbf A[/imath] is "skew-symmetric".
Can you see that this equivalent to equation [imath]S(\mathbf x, \mathbf y) = - S(\mathbf y, \mathbf x)[/imath]? The latter can be proven by looking at [imath]S(\mathbf u+\mathbf v , \mathbf u+\mathbf v)[/imath].