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In this paper, the singular-value decomposition theory of complex matrices isexplored to study constantly curved 2-spheres minimal in both $\mathbb{C}P^n$and the hyperquadric of $\mathbb{C}P^n$. The moduli space of all thosenoncongruent ones is introduced, which can be described by certain complexsymmetric matrices modulo an appropriate group action. Using this description,many examples, such as constantly curved holomorphic 2-spheres of higherdegree, nonhomogenous minimal 2-spheres of constant curvature, etc., areconstructed. Uniqueness is proven for the totally real constantly curved2-sphere minimal in both the hyperquadric and $\mathbb{C}P^n$.

Structure of minimal 2-spheres of constant curvature in the complex hyperquadric