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

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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.

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

Article copyright:
© Copyright 1974
American Mathematical Society