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/Summary

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