Characterization and properties of (Pσ, Q) symmetric and co-symmetric matrices

Abstract

Abstract Let P ∈ ℂmxm and Q ∈ ℂn×n be invertible matrices partitioned as P = [P0 P1 · · · Pk−1] and Q = [Q0 Q1 · · · Qk−1], with P ℓ ∈ ℂm×mℓ and Qℓ ∈ ℂn×nℓ , 0 ≤ ℓ ≤ k − 1. Partition P−1 and Q−1 as where P̂ℓ ∈ ℂmℓ ×m, Q̂ℓ ∈ ℂnℓ×n , P̂ℓPm = δℓmImℓ , and Q̂ℓQm = δℓmInℓ , 0 ≤ ℓ, m ≤ k − 1. Let Zk = {0, 1, . . . , k − 1}. We study matrices A = Pσ(ℓ)FℓQℓ and B = QℓGℓPσ(ℓ), where σ : Zk → Zk. Special cases: A = and B = , where Aℓ ∈ ℂd1×d2 and Bℓ ∈ ℂd2×d1, 0 ≤ ℓ ≤ k − 1.