Let Ωn denote the set of all nxn doubly stochastic matrices. A matrix B ∈ Ωn is said to be a star matrix if per(αB + (1 − α)A) ≤ α perB + (1 − α)perA, for all A ∈ Ωn and for all α ∈ [0, 1]. In this paper we derive a necessary condition for a star matrix to be in Ωn, and a partial proof of the star conjecture: The direct sum of two star matrices is a star matrix.
S. Maria Arulraj and Dr. Somasundaram K., “Star Matrices: Properties And Conjectures”, Applied Mathematics E-Notes, vol. 7, pp. 42–49, 2007.