Complex Hadamard Matrices contained in a Bose–Mesner algebra
Acomplex Hadamard matrix is a square matrix H with complex entries of absolute value 1 satisfying HH* = nI, where * stands for the Hermitian transpose and I is the identity matrix of order n. In this paper, we first determine the image of a certain rational map from the d-dimensional complex project...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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De Gruyter
2015-05-01
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| Series: | Special Matrices |
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| Online Access: | http://www.degruyter.com/view/j/spma.2015.3.issue-1/spma-2015-0009/spma-2015-0009.xml?format=INT |
| _version_ | 1818364764554788864 |
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| author | Ikuta Takuya Munemasa Akihiro |
| author_facet | Ikuta Takuya Munemasa Akihiro |
| author_sort | Ikuta Takuya |
| collection | DOAJ |
| description | Acomplex Hadamard matrix is a square matrix H with complex entries of absolute value 1 satisfying
HH* = nI, where * stands for the Hermitian transpose and I is the identity matrix of order n. In this paper,
we first determine the image of a certain rational map from the d-dimensional complex projective space to
Cd(d+1)/2. Applying this result with d = 3, we give constructions of complex Hadamard matrices, and more
generally, type-II matrices, in the Bose–Mesner algebra of a certain 3-class symmetric association scheme.
In particular, we recover the complex Hadamard matrices of order 15 found by Ada Chan. We compute the
Haagerup sets to show inequivalence of resulting type-II matrices, and determine the Nomura algebras to
show that the resulting matrices are not decomposable into generalized tensor products. |
| first_indexed | 2024-12-13T22:09:33Z |
| format | Article |
| id | doaj.art-40c1b0edd41247e4ba215714424480b2 |
| institution | Directory Open Access Journal |
| issn | 2300-7451 |
| language | English |
| last_indexed | 2024-12-13T22:09:33Z |
| publishDate | 2015-05-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Special Matrices |
| spelling | doaj.art-40c1b0edd41247e4ba215714424480b22022-12-21T23:29:45ZengDe GruyterSpecial Matrices2300-74512015-05-013110.1515/spma-2015-0009spma-2015-0009Complex Hadamard Matrices contained in a Bose–Mesner algebraIkuta Takuya0Munemasa Akihiro1Kobe Gakuin University, Kobe, 650-8586, JapanTohoku University, Sendai, 980-8579, JapanAcomplex Hadamard matrix is a square matrix H with complex entries of absolute value 1 satisfying HH* = nI, where * stands for the Hermitian transpose and I is the identity matrix of order n. In this paper, we first determine the image of a certain rational map from the d-dimensional complex projective space to Cd(d+1)/2. Applying this result with d = 3, we give constructions of complex Hadamard matrices, and more generally, type-II matrices, in the Bose–Mesner algebra of a certain 3-class symmetric association scheme. In particular, we recover the complex Hadamard matrices of order 15 found by Ada Chan. We compute the Haagerup sets to show inequivalence of resulting type-II matrices, and determine the Nomura algebras to show that the resulting matrices are not decomposable into generalized tensor products.http://www.degruyter.com/view/j/spma.2015.3.issue-1/spma-2015-0009/spma-2015-0009.xml?format=INTassociation scheme complex Hadamard matrix type-II matrix05E30; 05B34 |
| spellingShingle | Ikuta Takuya Munemasa Akihiro Complex Hadamard Matrices contained in a Bose–Mesner algebra Special Matrices association scheme complex Hadamard matrix type-II matrix 05E30; 05B34 |
| title | Complex Hadamard Matrices contained in a Bose–Mesner algebra |
| title_full | Complex Hadamard Matrices contained in a Bose–Mesner algebra |
| title_fullStr | Complex Hadamard Matrices contained in a Bose–Mesner algebra |
| title_full_unstemmed | Complex Hadamard Matrices contained in a Bose–Mesner algebra |
| title_short | Complex Hadamard Matrices contained in a Bose–Mesner algebra |
| title_sort | complex hadamard matrices contained in a bose mesner algebra |
| topic | association scheme complex Hadamard matrix type-II matrix 05E30; 05B34 |
| url | http://www.degruyter.com/view/j/spma.2015.3.issue-1/spma-2015-0009/spma-2015-0009.xml?format=INT |
| work_keys_str_mv | AT ikutatakuya complexhadamardmatricescontainedinabosemesneralgebra AT munemasaakihiro complexhadamardmatricescontainedinabosemesneralgebra |