 Library Home /
 Search Collections /
 Open Collections /
 Browse Collections /
 UBC Theses and Dissertations /
 Group matrices
Open Collections
UBC Theses and Dissertations
UBC Theses and Dissertations
Group matrices Iwata, William Takashi
Abstract
A new proof is given of Newman and Taussky's result: if A is a unimodular integral n x n matrix such that A′A is a circulant, then A = QC where Q is a generalized permutation matrix and C is a circulant. A similar result is proved for unimodular integral skew circulants. Certain additional new results are obtained, the most interesting of which are: 1) Given any nonsingular group matrix A there exist unique real group matrices U and H such that U is orthogonal and H is positive definite and A = UH; 2) If A is any unimodular integral circulant, then integers k and s exist such that A′ = P(k)A and P(s)A is symmetric, where P is the companion matrix of the polynomial xⁿ1. Finally, all the n x n positive definite integral and unimodular skew circulants are determined for values of n ≤ 6: they are shown to be trivial for n = 1,2,3 and are explicitly described for n = 4,5,6.
Item Metadata
Title  Group matrices 
Creator  Iwata, William Takashi 
Publisher  University of British Columbia 
Date Issued  1965 
Description 
A new proof is given of Newman and Taussky's result: if A is a unimodular integral n x n matrix such that A′A is a circulant, then A = QC where Q is a generalized permutation matrix and C is a circulant. A similar result is proved for unimodular integral skew circulants.
Certain additional new results are obtained, the most interesting of which are: 1) Given any nonsingular group matrix A there exist unique real group matrices U and H such that U is orthogonal and H is positive definite and A = UH; 2) If A is any unimodular integral circulant, then integers k and s exist such that A′ = P(k)A and P(s)A is symmetric, where P is the companion matrix of the polynomial xⁿ1.
Finally, all the n x n positive definite integral and unimodular skew circulants are determined for values of n ≤ 6: they are shown to be trivial for n = 1,2,3 and are explicitly described for n = 4,5,6.

Subject  Matrices 
Genre  Thesis/Dissertation 
Type  Text 
Language  eng 
Date Available  20111014 
Provider  Vancouver : University of British Columbia Library 
Rights  For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. 
DOI  10.14288/1.0080541 
URI  
Degree  Master of Arts  MA 
Program  Mathematics 
Affiliation  Science, Faculty of; Mathematics, Department of 
Degree Grantor  University of British Columbia 
Campus  UBCV 
Scholarly Level  Graduate 
Aggregated Source Repository  DSpace 
Item Media
Item Citations and Data
License
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.