On the singular value decomposition over finite fields and orbits of GU×GU
The singular value decomposition of a complex matrix is a fundamental concept in linear algebra and has proved extremely useful in many subjects. It is less clear what the situation is over a finite field. In this paper, we classify the orbits of GUm(q)×GUn(q) on Mm×n(q2) (which is the analog of the...
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| Published in | Indagationes mathematicae Vol. 32; no. 5; pp. 1083 - 1094 |
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| Format | Journal Article |
| Language | English |
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Elsevier B.V
01.09.2021
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| Abstract | The singular value decomposition of a complex matrix is a fundamental concept in linear algebra and has proved extremely useful in many subjects. It is less clear what the situation is over a finite field. In this paper, we classify the orbits of GUm(q)×GUn(q) on Mm×n(q2) (which is the analog of the singular value decomposition). The proof involves Kronecker’s theory of pencils and the Lang–Steinberg theorem for algebraic groups. Besides the motivation mentioned above, this problem came up in a recent paper of Guralnick et al. (2020) where a concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups was studied and bounds on the number of orbits was needed. A consequence of this work determines possible pairs of Jordan forms for nilpotent matrices of the form AA∗ and A∗A over a finite field and AA⊤ and A⊤A over arbitrary fields. |
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| AbstractList | The singular value decomposition of a complex matrix is a fundamental concept in linear algebra and has proved extremely useful in many subjects. It is less clear what the situation is over a finite field. In this paper, we classify the orbits of GUm(q)×GUn(q) on Mm×n(q2) (which is the analog of the singular value decomposition). The proof involves Kronecker’s theory of pencils and the Lang–Steinberg theorem for algebraic groups. Besides the motivation mentioned above, this problem came up in a recent paper of Guralnick et al. (2020) where a concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups was studied and bounds on the number of orbits was needed. A consequence of this work determines possible pairs of Jordan forms for nilpotent matrices of the form AA∗ and A∗A over a finite field and AA⊤ and A⊤A over arbitrary fields. |
| Author | Guralnick, Robert M. |
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| Cites_doi | 10.1080/00029890.1996.12004788 10.1007/BFb0081546 10.1016/0021-8693(82)90105-3 10.1016/j.laa.2010.03.022 10.1007/s00026-018-0390-4 10.2140/pjm.1959.9.893 10.1090/S0894-0347-97-00219-1 10.1007/BF01403155 |
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| Title | On the singular value decomposition over finite fields and orbits of GU×GU |
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