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Consider $ {x_k}_{k = 1}^K \in \mathbb{C}^{N \times 1}$ a set of $K$ complex vector variables of length $N$. I am interested in finding the existence of a solution of the following quadratic set of equations:

$ \sum_{k \neq i}^K x_k^HQ_{ik}x_k = x_i^HQ_{ii}x_i \quad i = 1, \ldots K$

$ x_i^Hx_i = 1 \quad i = 1, \ldots K$

Where $Q_{ij} \quad i=1, \ldots , K \quad j=1, \ldots , K$ are assumed to be hermitian and semidefinite positive. I do not have any clue of the direction to take, any comment is highly welcomed.

-----------Edition-------------

Of course, $Q_{ij} \in \mathbb{C}^{N \times N}$ and $N > 1$

edited Feb 12 at 8:05

asked Feb 11 at 17:47

mikitov
105●7

in general, the solution does not exist; take for example $Q_{ij}=\delta_{ij}$, then your equations imply 0=1. – Carlo Beenakker Feb 11 at 20:51

Q_{ij} are complex hermitian matrices. – mikitov Feb 11 at 23:25

isn't the unit matrix $\in \mathbb{C}^{N\times N}$? – Carlo Beenakker Feb 12 at 8:24

I would like to know the conditions so that there is a solution to the equation system. – mikitov Feb 12 at 11:20

1

I'm not sure if it helps, but there is a simple semidefinite program relaxation of your system which could provide certificates of infeasibility in some cases. Perhaps looking at the dual would even give a simple sufficient condition for infeasibility. Unfortunately this does not answer the question of when a solution does exist, except in cases where you get lucky and a solution of the relaxation is feasible for the given problem. Also it won't provide an analytically-checkable sufficient condition for feasibility. – Noah Stein Feb 12 at 16:00

Solution Existence of a System of Complex Quadratic Equations

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