Multilevel matrices with involutory symmetries and skew symmetries

Abstract

Let n = n 1 n 2 ⋯ n k where k > 1 and n 1, … , n k are integers >1. For 1 ⩽ i ⩽ k, let p i = ∏ j = 1 i - 1 n j and q i = ∏ j = i + 1 k n j , and suppose that U i ∈ C n i × n i is a nontrivial involution; i.e., U i = U i - 1 ≠ ± I n i . Let R i = I p i ⊗ U i ⊗ I q i , 1 ⩽ i ⩽ k, and denote R = ( R 1, … , R k ). If μ ∈ { 0, 1, l… , 2 k−1 }, let μ = ∑ i = 1 k ℓ i μ 2 i - 1 be its binary expansion. We say that A ∈ C n × n is ( R, μ)-symmetric if R i AR i = ( - 1 ) ℓ i μ A , 1 ⩽ i ⩽ k; thus, we are considering matrices with k levels of block structure and an involutory symmetry or skew symmetry at each level. We characterize the class of all ( R, μ)-symmetric matrices and study their properties. The theory divides into two parts corresponding to μ = 0 and μ ≠ 0. Problems involving an ( R, 0)-symmetric matrix split into the corresponding problems for 2 k−1 matrices with orders summing to n, while problems involving an ( R, μ)-symmetric matrix with μ ≠ 0 split into the corresponding problems for 2 k−1 −1 matrices with orders summing to n. The latter is also true of A = B + C where B is ( R, 0)-symmetric and C is R, μ)-symmetric with μ ≠ 0.