I am reading Scott Aaronson, Is Quantum Mechanics An Island In Theoryspace? In Section 2 Other p-Norms, he tries to prove for $p>2$, the p-norm preserving matrix is generalized diagonal. Namely, for matrix $A=[a_{j,k}]_{j,k=1}^n$, $$\sum_{j=1}^n |x_j|^p=\sum_{j=1}^n\Big|\sum_{k=1}^n a_{j,k}x_k\Big|^p\,\,\forall (x_j)_{j=1}^n \implies [a_{j,k}] \text{ is a generalized diagonal matrix}$$ I am not sure what exactly a generalized diagonal matrix is. I first guessed it was a permutation matrix with its $1$'s replaced by another number, then I surmised it was a permutation matrix with its $1$'s replaced by $\pm1$'s. He proves this for separate cases. I am having trouble understanding his step for when $p$ is an odd positive integer.
He "claim that, so long as $x_1,\dots,x_n$ are nonnegative, the entries of $Ax$ never change sign." First I do not quite understand what he means by changing signs. I suppose he means the sign of each number $\sum_{j=1}^n a{j,k}x^{(1)}_k$ is the same as that of number $\sum_{j=1}^n a{j,k}x^{(2)}_k$ $\forall j$ but different vectors $x^{(1)}_k\neq x^{(2)}_k$. I am do not understand where $(s_j)_j$ comes from. It does not look like it is just $s_j=\text{sgn}y_j$. I would appreciate if someone can elucidate.