Hodge star/ Technical question

Question

If we have an equation that looks like $$H=Y$$ and we want to multiply $H$ by either $ReM_{IJ}$ or $ImM_{IJ}$ where $M_{IJ}$ is a complex matrix. But the thing is that $$Y=\star(...)$$ where $\star$ is hodge star and (...) is set of complex functions and other numerical stuff, my question is technical here, say we decide to multiply H by $ReM_{IJ}$ can we move $ReM_{IJ}$ into the parenthesis and jump over the $\star$? That is to say $$ReM_{IJ}H=\star(ReM_{IJ} ...)$$ or this is absolutely wrong and we should keep $ReN_{IJ}$ outside the $\star$? That is to say $$ReM_{IJ}H=ReM_{IJ}\star( ...)$$

Answer 1

After some discussion, it seems like your question is actually the following:

Let $\omega$ be a differential form and $c \in \mathbb{R}$. Is $\star(c\omega) = c(\star\omega)$?

The answer to this is yes. The Hodge dual is real-linear, in fact it is linear over real-valued functions, i.e. $\star(f\omega) = f(\star\omega)$.

If you want to bring out a complex number or complex-valued function, then it depends on your definition of $\star$; in particular, whether it is complex linear or conjugate linear.