  Necessary and sufficient conditions for the existence of an entropy structure
for certain classes of cross-diffusion systems with diffusion matrix $A(u)$ are
derived, based on results from matrix factorization. The entropy structure is
important in the analysis for such equations since $A(u)$ is typically neither
symmetric nor positive definite. In particular, the normal ellipticity of
$A(u)$ for all $u$ and the symmetry of the Onsager matrix implies its positive
definiteness and hence an entropy structure. If $A$ is constant or nearly
constant in a certain sense, the existence of an entropy structure is
equivalent to the normal ellipticity of $A$. Several applications and examples
are presented, including the $n$-species population model of Shigesada,
Kawasaki, and Teramoto, a volume-filling model, and a fluid mixture model with
partial pressure gradients. Furthermore, the normal elipticity of these models
is investigated and some extensions are discussed.
