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 xⁿ-1.
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. / Science, Faculty of / Mathematics, Department of / Graduate
| Identifer | oai:union.ndltd.org:UBC/oai:circle.library.ubc.ca:2429/38015 |
| Date | January 1965 |
| Creators | Iwata, William Takashi |
| Publisher | University of British Columbia |
| Source Sets | University of British Columbia |
| Language | English |
| Detected Language | English |
| Type | Text, Thesis/Dissertation |
| Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
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