On the singular value decomposition over finite fields and orbits of GU×GU
| Autor: | Robert M. Guralnick |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Pure mathematics
General Mathematics 010102 general mathematics 010103 numerical & computational mathematics 01 natural sciences Unitary state Nilpotent matrix symbols.namesake Finite field Character (mathematics) Kronecker delta Singular value decomposition Linear algebra symbols 0101 mathematics Algebraic number Mathematics |
| Zdroj: | Indagationes Mathematicae. 32:1083-1094 |
| ISSN: | 0019-3577 |
| DOI: | 10.1016/j.indag.2021.01.006 |
| Popis: | 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 GU m ( q ) × GU n ( q ) on M m × n ( q 2 ) (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 A A ∗ and A ∗ A over a finite field and A A ⊤ and A ⊤ A over arbitrary fields. |
| Databáze: | OpenAIRE |
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