For M different hermitian forms $h_i(\vec{c}_0,\vec{c}_1)$, where $\vec{c}_j$ are complex vectors in $\mathbb{C}^N$, I want to calculate $$\min_{\vec{c} \in \mathbb{C}^N\backslash\{\vec{0}\}} \frac{\sum_{i=1}^M h_i^2(\vec{c},\vec{c})}{|\vec{c}|^4}.$$ I have additional information, namely knowledge in a basis of $$\sum_{i=1}^M h_i(\vec{e}_i,\vec{e}_j) h_i(\vec{e}_k,\vec{e}_l)$$ for all combinations of $i,j,k,l$, where $\{\vec{e}_1,\dots,\vec{e}_{2 N}\}$ is a basis of $\mathbb{C}^N$ over the reals. How can I solve the optimization problem analytically?

To me this looks like a Tensor of order 4, with a "Rayleigh like" quotient. Is it possible to formulate this in some way as a higher dimensional eigenvalue problem (like we can do for matrices)?

Thank you

  • $\begingroup$ Do you mind explaining your given information more clearly? Is that sum_{i,j,k,l} h(v_i,v_j)h(v_k,v_l) ? $\endgroup$ – MeowBlingBling Dec 13 '19 at 4:42
  • $\begingroup$ Sry if I was not clear. The sum is over the hermitian forms, each represents a contribution of a particle to some energy like term. So I need to add them up for the total contribution. $\endgroup$ – Nonabelian Dec 13 '19 at 12:08

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