A Linear-algebraic Proof of Hilbert's Ternary Quartic Theorem

Hilbert's ternary quartic theorem states that every nonnegative degree 4 homogeneous polynomial in three variables can be written as a sum of three squares of homogeneous quadratic polynomials. We give a linear-algebraic approach to Hilbert's theorem by showing that a structured cone of positive semidefinite matrices is generated by rank 1 elements.