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Mordred, I have a technical question here.... my work is a bit more complicated and requires a bit more depth than I am used to  in the case of a projective space, the identity P^2 = P is said to hold  what does it mean?
In my case I am looking at, I have an antisymmetric matrix that requires to be squared in the projected space to yield the identity/unity. The projective space looks like
[math]P = \frac{\mathbf{I} + n \cdot \sigma}{2} = \psi><\psi[/math]The square of the Pauli matrix should yield an identity [math](n \cdot \sigma)^2 = \mathbf{I}[/math] (unit vectors naturally square into unity). What is the square of the dyad in such a case?
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[math]\rho^2=\rho[/math] is the projector condition [math]\rho[/rho is only a projector if and only if [math]\rho^2=\rho[/math]
wiki explains it a bit better
In operator language, a density operator is a positive semidefinite, Hermitian operator of trace 1 acting on the state space.^{[7]} A density operator describes a pure state if it is a rank one projection. Equivalently, a density operator ρ describes a pure state if and only if
 ρ=ρ2{\displaystyle \rho =\rho ^{2}},
i.e. the state is idempotent. This is true regardless of whether H is finitedimensional or not.
https://en.wikipedia.org/wiki/Density_matrix