An analog of Hölder's inequality for the spectral radius of Hadamard products

Abstract

We prove new inequalities related to the spectral radius ρ of Hadamard products (denoted by ◦ ) of complex matrices. Let p, q ∈ [1 , ∞ ] satisfy 1/p + 1/q = 1, we show an analog of Hölder’s inequality on the space of n × n complex matrices ρ ( A ◦ B ) ≤ ρ ( | A |^( ◦ p) ) ^(1/p) ρ ( | B |^( ◦ q) ) ^(1/q) for all A, B ∈ C n × n , where |·| denotes entry-wise absolute values, and ( · ) ^ (◦ p) represents the entry-wise Hadamard power. We derive a sharper inequality for the special case p = q = 2. Given A, B ∈ C ^(n × n) , for some β ∈ (0 , 1] depending on A and B , ρ ( A ◦ B ) ≤ βρ ( | A ◦ A | ) ^(1/2) ρ ( | B ◦ B | )^ (1/2) . Analysis for another special case p = 1 and q = ∞ is also included.