Approximating complex perplectic matrices by finite products from a finite generating set

Author

Aaron Pagaygay

Date of Publication

2021

Document Type

Master's Thesis

Degree Name

Master of Science in Mathematics

Subject Categories

Mathematics

College

College of Science

Department/Unit

Mathematics and Statistics Department

Thesis Advisor

John Vincent S. Morales

Defense Panel Chair

Arlene A. Pascasio

Defense Panel Member

Jose Tristan F. Reyes
Daryl Q. Granario

Abstract (English)

For positive integer n, let 𝓜_n(ℂ) denote the set of all n x n matrices over ℂ. We say a matrix A in 𝓜_n(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 𝓖_m = {A₁A₂...A_k | 0 ≤ k ≤ m and A₁,...,A_k ∈ 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<∞𝓖_m. 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.

Abstract Format

html

Language

English

Format

Electronic

Physical Description

[43 leaves]

Keywords

Matrices; Lie algebras

Recommended Citation

Pagaygay, A. (2021). Approximating complex perplectic matrices by finite products from a finite generating set. Retrieved from https://animorepository.dlsu.edu.ph/etdm_math/2

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Embargo Period

9-11-2021