Suppose a group $G$ acts on a set $X$. I know what is meant by "an orbit of a point $x\in X$".
For $Y\subset X$, what does "$Y$ is an orbit" mean? Does it mean "for some $x\in X$, $Y$ is an orbit of $x$" ?
Consider a right action $(P, A) \mapsto {}^tPA\bar P$ of $\mathit{GL}_n(\mathbb{C})$ on $M_n(\mathbb{C})$. A textbook at hand says a subset $\{A\in M_n(\mathbb{C})\mid A\ \text{is a positive-definite Hermitian matrix}\}\subset M_n(\mathbb{C})$ is an orbit. Why is this true?
Edit: Since the word something seemed confusing, I replaced it with $Y$.