Similitudes and the \(\ell_ 1\)-norm (Q1923200)

Let \(C= \{c\in \mathbb{R}^n: \sum^n_{i=1} c_i= 0\}\), and let \(\alpha>0\). The author characterizes the matrices \(Q:C \to C\) satisfying \(|cQ |= \alpha |c |\) for all \(c \in C\), where \(|\cdot |\) denotes the \(\ell_1\)-norm. They are precisely of the form \(Q= \pm \alpha P+F\), where \(P\) is a permutation matrix, and \(F\) has all rows identical. In terms of the coefficient of ergodicity \({\mathcal T} (A) = \max \{|cA |/ |c |: c\in C,\;c \neq 0\}\), the characterization is that \({\mathcal T} (AQ) = {\mathcal T} (QA) = \text{const.} {\mathcal T} (A)\) for all stochastic matrices \(A\).