Some spectral norm inequalities on Hadamard products of nonnegative matrices.

Let A and B be nonnegative square matrices of the same order. Denote by ‖ ⋅ ‖ and ρ ( ⋅ ) the spectral norm and the spectral radius respectively. We prove the following inequalities: ‖ A ∘ B ‖ ≤ ‖ A ∘ A ‖ 1 2 ‖ B ∘ B ‖ 1 2 ; ‖ A ∘ B ‖ ≤ ρ 1 2 ( A T B ∘ B T A ) ≤ ρ 1 2 ( A T B ∘ A T B ) ≤ ρ ( A T B ) , where ∘ denotes the Hadamard product. This interpolates the inequality ‖ A ∘ B ‖ ≤ ρ ( A T B ) due to Huang. Some spectral norm inequalities for an arbitrarily finite number of nonnegative square matrices are also obtained, which refine some other results of Huang. [ABSTRACT FROM AUTHOR]