This paper studies lower and upper bounds for the number of vertices of the polytope of stochastic tensors (i.e. triply stochastic arrays of dimension n). By using known results on polytopes (i.e. the Upper and Lower Bound Theorems), we present some new lower and upper bounds. We show that the new upper bound is tighter than the one recently obtained by Chang et al. [Ann Funct Anal. 2016;7(3):386–393] and also sharper than the one in Linial and Luria’s [Discrete Comput Geom. 2014;51(1);161–170]. We demonstrate that the analog of the lower bound obtained in such a way, however, is no better than the existing ones.

This note shows the equivalence of two well-known inequalities: the Wielandt inequality and the Kantorovich inequality.

This is continuation of the recent work by Xu, Xu and Zhang [Revisiting Hua–Marcus–Bellman–Ando inequalities on contractive matrices, Linear Algebra Appl. 430 (2009), pp. 1499–1508] on contractive matrices. We study the relations of block matrices of Hua type, present some properties that the eigenvalues of Hua matrices possess, especially for the 2 × 2 block case, discuss the analogues for higher dimensions and estimate the closeness of two Hua matrices. At the end, we propose a conjecture on the eigenvalues of Hua matrices and an open problem on the symmetric functions of the eigenvalues of contractive matrices.