Perron number

In mathematics, a Perron number is an algebraic integer α which is real and greater than 1, but such that its conjugate elements are all less than α in absolute value. For example, the larger of the two roots of the irreducible polynomial is a Perron number.

Perron numbers are named after Oskar Perron; the Perron–Frobenius theorem asserts that, for a real square matrix with positive algebraic entries whose largest eigenvalue is greater than one, this eigenvalue is a Perron number. (Conversely, it was shown in the late 20th century by Douglas Lind that the Perron numbers are in fact the "closure" of the natural numbers, realized after only one iteration of taking spectral radii of aperiodic matrices, where entries are drawn from this set.) As a closely related case, the Perron number of a graph is defined to be the spectral radius of its adjacency matrix.

Any Pisot number or Salem number is a Perron number, as is the Mahler measure of a monic integer polynomial.

References

  • Borwein, Peter (2007). Computational Excursions in Analysis and Number Theory. Springer Verlag. p. 24. ISBN 978-0-387-95444-8.


Stub icon

This number theory–related article is a stub. You can help Wikipedia by adding missing information.

Stub icon

This graph theory-related article is a stub. You can help Wikipedia by adding missing information.

Content Disclaimer

Informasi ini disarikan dari Wikipedia dan disajikan kembali untuk tujuan edukasi. Konten tersedia di bawah lisensi CC BY-SA 3.0. Kami tidak bertanggung jawab atas ketidakakuratan data yang bersumber dari kontribusi publik tersebut.

  1. The information displayed on this website is sourced in part or in whole from Wikipedia and has been adapted for the purpose of restating it. We strive to provide accurate and relevant information, however:
  2. There is no guarantee of absolute accuracy. Wikipedia is an open, collaborative project that can be edited by anyone, so information is subject to change.
  3. It is not intended to constitute professional advice. The content displayed is for informational and educational purposes only. For important decisions (e.g., medical, legal, or financial), please consult a professional.
  4. Content copyright. Wikipedia is licensed under the Creative Commons Attribution-ShareAlike License (CC BY-SA). This means that content may be reused with appropriate attribution and shared under a similar license.
  5. Responsible use. Any risk arising from the use of information from this website is entirely the responsibility of the user.