A linear algebra approach to graded Frobenius algebras

If $A$ is a finite-dimensional algebra graded by a group $G$, and $\sigma \in G$, we define a variant of paratrophic matrix associated with $A$ and $\sigma$, and we use it to characterize the $\sigma$-graded Frobenius property for $A$. We discuss the invertibility of such paratrophic matrices, and then use them to check whether certain graded algebras are $\sigma$-graded Frobenius or (graded) symmetric. As an application, we uncover (graded) Frobenius and symmetric properties of Koszul duals of quantum polynomial algebras. We derive a structure result for $\sigma$-graded Frobenius algebras by only using linear algebra methods.