Rank solutions to the matrix equation alternate, over

Author:
Philip G. Buckhiester

Journal:
Trans. Amer. Math. Soc. **189** (1974), 201-209

MSC:
Primary 15A33

DOI:
https://doi.org/10.1090/S0002-9947-1974-0330196-0

MathSciNet review:
0330196

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let denote a finite field of characteristic two. Let denote an n-dimensional vector space over . An symmetric matrix *A* over is said to be an alternate matrix if *A* has zero diagonal. Let *A* be an alternate matrix over and let *C* be an symmetric matrix over . By using Albert's canonical forms for symmetric matrices over fields of characteristic two, the number of matrices *X* of rank *r* over such that is determined.

A symmetric bilinear form on is said to be alternating if , for each *x* in . Let *f* be such a bilinear form. A basis , for is said to be a symplectic basis for if and , for each *i*, . In determining the number , it is shown that a symplectic basis for any subspace of , can be extended to a symplectic basis for . Furthermore, the number of ways to make such an extension is determined.

**[1]**A. Adrian Albert,*Symmetric and alternate matrices in an arbitrary field. I*, Trans. Amer. Math. Soc.**43**(1938), no. 3, 386–436. MR**1501952**, https://doi.org/10.1090/S0002-9947-1938-1501952-6**[2]**J. V. Brawley and L. Carlitz,*Enumeration of matrices with prescribed row and column sums*, Linear Algebra and Appl.**6**(1973), 165–174. MR**0314869****[3]**Philip G. Buckhiester,*Gauss sums and the number of solutions to the matrix equation 𝑋𝐴𝑋^{𝑇}=0 over 𝐺𝐹 (2^{𝑦})*, Acta Arith.**23**(1973), 271–278. MR**0325644****[4]**L. Carlitz,*Representations by quadratic forms in a finite field*, Duke Math. J.**21**(1954), 123–137. MR**0059952****[5]**L. Carlitz,*The number of solutions of certain matric equations over a finite field*, Math. Nachr.**56**(1973), 105–109. MR**0325645**, https://doi.org/10.1002/mana.19730560109**[6]**Claude C. Chevalley,*The algebraic theory of spinors*, Columbia University Press, New York, 1954. MR**0060497****[7]**Dai Zong-duo,*On transitivity of subspaces in orthogonal geometry over fields of characteristic 2*, Chinese Math.–Acta**8**(1966), 569–584. MR**0209311****[8]**Leonard Eugene Dickson,*Linear groups: With an exposition of the Galois field theory*, with an introduction by W. Magnus, Dover Publications, Inc., New York, 1958. MR**0104735****[9]**John C. Perkins,*Rank 𝑟 solutions to the matrix equation 𝑋𝑋^{𝑇}=0 over a field of characteristic two*, Math. Nachr.**48**(1971), 69–76. MR**0294365**, https://doi.org/10.1002/mana.19710480106**[10]**John C. Perkins,*Gauss sums and the matrix equation 𝑋𝑋^{𝑇}=0 over fields of characteristic two*, Acta Arith.**19**(1971), 205–214. MR**0291189**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
15A33

Retrieve articles in all journals with MSC: 15A33

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1974-0330196-0

Article copyright:
© Copyright 1974
American Mathematical Society