  Let $F$ be an algebraically closed field of characteristic zero, and $G$ be a
finite abelian group. If $A=\oplus_{g\in G} A_g$ is a $G$-graded algebra, we
study degree-inverting involutions on $A$, i.e., involutions $*$ on $A$
satisfying $(A_g)^*\subseteq A_{g^{-1}}$, for all $g\in G$. We describe such
involutions for the full $n\times n$ matrix algebra over $F$ and for the
algebra of $n\times n$ upper triangular matrices.
