Date of Publication


Document Type

Master's Thesis

Degree Name

Master of Science in Mathematics

Subject Categories



College of Science


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


For positive integer n, let π“œn(β„‚) denote the set of all n x n matrices over β„‚. We say a matrix A in π“œn(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(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 π“œ2(β„‚). 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 βŸ¨π“–βŸ© = ⋃0≀m<βˆžπ“–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 βŸ¨π“–βŸ© 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.

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Physical Description

[43 leaves]


Matrices; Lie algebras

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