#### 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 *π*_{n}*(**β**)* denote the set of all *n x n* matrices over *β*. We say a matrix *A *in *π*_{n}*(**β**) *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*

*(x*

_{1}*U*

_{1}*+ x*

_{2}*U*

_{2}*+ x*

_{3}*U*where

_{3})*x*

_{1}*,x*

_{2}*,x*

*are real numbers such that*

_{3}*4x*

_{2}*x*

_{3}*-x*

_{1}

^{2}*> 0*and the matrices

*U*

_{1}*,U*

_{2}*,U*span the complex perplectic Lie algebra of order two. Using polar and LDL decompositions, we obtain the decomposition

_{3}*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 *π* denote a nonempty finite subset of *π*_{2}*(**β**)*. For each positive integer *m*, we define *π*_{m}* = {A _{1}A_{2}...A_{k} | 0 β€ k β€ m and A_{1}*

*,...,A*

_{k}*β*

*π*

*}*which is the set of all words in

*G*of length at most

*m*where word of length 0 is the identity. We abbreviate

*β¨πβ©*

*=*

*β*

_{0β€m<β}*π*

*. In this paper, we construct a fixed set*

_{m}*π*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

*β¨πβ©*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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9-11-2021