How could a describe a complex vector, in which the absolute value of the elements is all 1? It is not that the norm of the vector is 1, but the norm of all its elements.
For example:
$x = [ 1 e^{j20}; 1 e^{j30}; 1e^{j15}; 1e^{j2}]$
(1) $x^* x = 4$, where x* is the hermitian conjugate
But only condition (1) is not enough. Since we could have a vector
$x = [\sqrt{4} e^{j0} ; 0 ; 0; 0]$, where $x^*x = 4$.
So I need another condition to the constraint that each element's absolute value is 1. This constraint will be used in a lagrangian, so I must describe it neatly through matrix/vector algebra.
This is part of a dynamical system of the type: $\dot x = A x $
Any suggestions?
Can you use the "angles" ($\theta_i$) as your variables, and then write the rest of your formulas in terms of $e^{j\theta_i}$?
So instead of
$$x = [ 1 e^{j20}; 1 e^{j30}; 1e^{j15}; 1e^{j2}]$$
you would have
$$ \theta = [ 20; 30; 15; 2] $$
and if your function previously had been, say, $x_1 -2x_2^2$, it would now be $e^{j\theta_1} -2e^{2j\theta_2}$.