  We solve the Symmetrized Principal Minor Assignment Problem, that is we show
how to determine if for a given vector $v\in \mathbb{C}^{n}$ there is an
$n\times n$ matrix that has all $i\times i$ principal minors equal to $v_{i}$.
We use a special isomorphism (a non-linear change of coordinates to cycle-sums)
that simplifies computation and reveals hidden structure. We use the symmetries
that preserve symmetrized principal minors and cycle-sums to treat 3 cases:
symmetric, skew-symmetric and general square matrices. We describe the matrices
that have such symmetrized principal minors as well as the ideal of relations
among symmetrized principal minors / cycle-sums. We also connect the resulting
algebraic varieties of symmetrized principal minors to tangential and secant
varieties, and Eulerian polynomials.
