What is the manifold of x^H A x<=1, where A is a complex semi definite matrix, and x is a constant envelope complex vector？

[proki...@gmail.com]

Hi，

I’m fresh there, and I want to find the manifold of the question above. Actually, I checked the manifold list, the manifold x^H A x=1, is kind of like generalized stiefel manifold when x is a vector of real numbers. However，I do not know what it would be in the complex case and with the inequality constraints. Any ideas or tutorials would be appreciated! Thanks!

[BM]

Hello Prokins,

The manifold of interest would be spherecomplexfactory. To use this, we have to first use the varible z = B*x instead of x, where B is the matrix square root of A (which I would assume to be positive definite). The constraint in z would be z'*z = 1, which the complex sphere factory.  x is substituted by Binv*z. Looking carefully at the problem, it may be possible to avoid storing Binv (inverse of B explicitly).

Regards,

Bamdev