𝔖 Bobbio Scriptorium
✦   LIBER   ✦

Similarity to Symmetric Matrices over Finite Fields

✍ Scribed by Joel V. Brawley; Timothy C. Teitloff

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
299 KB
Volume
4
Category
Article
ISSN
1071-5797

No coin nor oath required. For personal study only.

✦ Synopsis

It has been known for some time that every polynomial with coefficients from a finite field is the minimum polynomial of a symmetric matrix with entries from the same field. What have remained unknown, however, are the possible sizes for the symmetric matrices with a specified minimum polynomial and, in particular, the least possible size. In this paper we answer these questions using only the prime factorization of the given polynomial. Closely related is the question of whether or not a given matrix over a finite field is similar to a symmetric matrix over that field. Although partial results on that question have been published before, this paper contains a complete characterization.

πŸ“œ SIMILAR VOLUMES

q-Rook Polynomials and Matrices over Fin
q-Rook Polynomials and Matrices over Finite Fields
✍ James Haglund πŸ“‚ Article πŸ“… 1998 πŸ› Elsevier Science 🌐 English βš– 337 KB

Connections between q-rook polynomials and matrices over finite fields are exploited to derive a new statistic for Garsia and Remmel's q-hit polynomial. Both this new statistic mat and another statistic for the q-hit polynomial recently introduced by Dworkin are shown to induce different multiset Ma

Cyclic Matrices in Classical Groups over
Cyclic Matrices in Classical Groups over Finite Fields
✍ Peter M. Neumann; Cheryl E. Praeger πŸ“‚ Article πŸ“… 2000 πŸ› Elsevier Science 🌐 English βš– 362 KB

to helmut wielandt for his 90th birthday with much respect and many congratulations , where m X t is its minimal polynomial and c X t is its characteristic polynomial det tI -X . This condition is equivalent to requiring the vector space F d of 1 Γ— d row vectors over F to be cyclic as an F X -modul

Invariants of 2Γ—2-Matrices over Finite F
Invariants of 2Γ—2-Matrices over Finite Fields
✍ Larry Smith πŸ“‚ Article πŸ“… 2002 πŸ› Elsevier Science 🌐 English βš– 114 KB

Let F q be the finite field with q elements, q ΒΌ p n ; p 2 N a prime, and Mat 2:2 Γ°F q Þ the vector space of 2 Γ‚ 2-matrices over F. The group GLΓ°2; FÞ acts on Mat 2;2 Γ°F q Þ by conjugation. In this note, we determine the invariants of this action. In contrast to the case of an infinite field, where

The rank of sparse random matrices over
The rank of sparse random matrices over finite fields
✍ Johannes BlΓΆmer; Richard Karp; Emo Welzl πŸ“‚ Article πŸ“… 1997 πŸ› John Wiley and Sons 🌐 English βš– 182 KB πŸ‘ 2 views

Let M be a random n = n -matrix over GF q such that for each entry M in i j w x Ž . M and for each nonzero field element ␣ the probability Pr M s ␣ is pr q y 1 , where i j ## Ž . p slog n y c rn and c is an arbitrary but fixed positive constant. The probability for a Ž . matrix entry to be zero