Multilevel matrices with involutory symmetries and skew symmetries

Let n = n1n2 ⋯ nk where k > 1 and n1, … , nk are integers >1. For 1 ⩽ i ⩽ k, let pi=∏j=1i-1nj and qi=∏j=i+1knj, and suppose that Ui∈Cni×ni is a nontrivial involution; i.e., Ui=Ui-1≠±Ini. Let Ri=Ipi⊗Ui⊗Iqi, 1 ⩽ i ⩽ k, and denote R = (R1, … , Rk). If μ ∈ { 0, 1, l… , 2k−1}, let μ=∑i=1kℓiμ2i-1 be its binary expansion. We say that A∈Cn×n is (R, μ)-symmetric if RiARi=(-1)ℓiμA, 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 2k−1 matrices with orders summing to n, while problems involving an (R, μ)-symmetric matrix with μ ≠ 0 split into the corresponding problems for 2k−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.