Determinantal representations of smooth cubic surfaces

For every smooth (irreducible) cubic surface $S$ we give an explicit construction of a representative for each of the 72 equivalence classes of determinantal representations. Equivalence classes (under ${\text{GL}}_3\times {\text{GL}}_3$ action by left and right multiplication) of determinantal representations are in one to one correspondence with the sets of six mutually skew lines on $S$ and with the 72 (two-dimensional) linear systems of twisted cubic curves on $S$. Moreover, if a determinantal representation $M$ corresponds to lines $(a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_6)$ then its transpose $M^t$ corresponds to lines $(b_1,\mathrm{.}\mathrm{.}\mathrm{.},b_6)$ which together form a Schl"{a}fli’s double-six $a_1… a_6 \choose b_1… b_6$. We also discuss the existence of self-adjoint and definite determinantal representation for smooth real cubic surfaces. The number of these representations depends on the Segre type $F_i$. We show that a surface of type $F_i$, $i=1,2,3,4$ has exactly $2(i-1)$ nonequivalent self-adjoint determinantal representations none of which is definite, while a surface of type $F_5$ has 24 nonequivalent self-adjoint determinantal representations, 16 of which are definite.