Approximating complex perplectic matrices by finite products from a finite generating set - Pagaygay, Aaron (Animo Repository, 2021)
For positive integer n, let 𝓜n(ℂ) denote the set of all n x n matrices over ℂ. We say a matrix A in 𝓜n(ℂ) is a complex perplectic matrix whenever A is invertible and A-1=JA*J such that J is the matrix with 1s on the skew-diagonal and 0s everywhere else, and A* is the conjugate-transpose of A. The matrix A is said to be skew-perHermitian whenever -A=JA*J. It turns out that the set of all complex perplectic matrices forms a matrix Lie group whose Lie algebra is the set of all skew-perHermitian matrices. Now, consider an arbitrary 2 x 2 perplectic matrix B of the form B=exp(x1U1 + x2U2 + x3U3) where x1,x2,x3 are real numbers such that 4x2x3-x12 > 0 and the matrices U1,U2,U3 span the complex perplectic Lie algebra of order two. Using polar and LDL decompositions, we obtain the decomposition B = ULDL* such that U is unitary, L is lower triangular, and D is diagonal. We show that U, L, D are all complex perplectic and have determinant 1. Let 𝓖 denote a nonempty finite subset of 𝓜2(ℂ). For each positive integer m, we define 𝓖m = {A1A2...Ak | 0 ≤ k ≤ m and A1,...,Ak ∈ 𝓖} which is the set of all words in G of length at most m where word of length 0 is the identity. We abbreviate ⟨𝓖⟩ = ⋃0≤m<∞𝓖m. In this paper, we construct a fixed set 𝓖 consisting of finitely many complex perplectic matrices that is closed under taking inverses. We show that U, L, D above can be approximated by some words in ⟨𝓖⟩ via the Hilbert-Schmidt norm. This leads to an approximation of the matrix B. Our results serve as initial steps towards establishing an analogue of the Solovay-Kitaev theorem on special complex perplectic group of order two.