I am interested in characterizing the definiteness of the following bilinear form over the torus:
$$\langle x, A x \rangle$$ $$A = A^T, A \in \mathbb{R}^{2n \times 2n}$$ Where the inner product denotes the standard dot product. By torus, I mean vectors of the form: $$x \in \mathbb{R}^{2n}, x = \begin{bmatrix} v_1\\v_2\\...\\v_i \end{bmatrix}, v_i \in \mathbb{R}^2, ||v_i||_2^2 = 1$$
Clearly, a sufficient condition for A to be positive-semidefinite over the Torus is for it to be positive semi-definite over $\mathbb{R}^{2n}$, but I don't believe this is a necessary condition.
What I've tried so far is to block decompose A. For example, in the 2x2 case this would look like: $$\langle x, A x \rangle = v_1^T A_{11} v_1 + v_2^T A_{22} v_2 + 2 v_2^T A_{21} v_1, \quad ||v_1||_2^2 = 1, ||v_2||_2^2 = 1 $$ Where $A_{ij}$ is the appropriate block matrix. The definiteness of A over the torus would then be equivalent to the definiteness of this expression. However, it's not obvious that the definiteness of this expression is any easier to characterize.
Are there simple necessary and sufficient conditions for A to be definite over the Torus (such as conditions on the eigenvalues of the block matrices)? I've done a fairly extensive search myself but haven't been able to find anything. Any help would be greatly appreciated.