On the spectral norm of a doubly stochastic matrix and level-\(k\) circulant matrix (Q6548446)

The Birkhoff theorem says that every doubly stochastic matrix can be written as a convex combination of permutation matrices. This observation immediately yields that the spectral norm of a doubly stochastic matrix is one, since the matrix already has an eigenvalue \(1\). The authors use this fact to generalize some results to circulant matrices, \(r\)-circulant matrices and level-\(k\) circulant matrices.