Suppose \(\mathbb{K}\) is either \(\mathbb{C}\) or \(\mathbb{R}\). The non-increasing rearrangement of a vector \(x\in \mathbb{K}^N\) is a vector \(x^*\in \mathbb{K}^N\) with \(x^*_i=\lvert x_{\pi(i)}\rvert\) for some permutation \(\pi\in\Sym(N)\) and with the property
\[ x^*_1\ge X^*_2}\ge \cdots \ge x^*_N. \]
Remarks