Rank solutions to the matrix equation alternate, over
Author:
Philip G. Buckhiester
Journal:
Trans. Amer. Math. Soc. 189 (1974), 201209
MSC:
Primary 15A33
MathSciNet review:
0330196
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Abstract: Let denote a finite field of characteristic two. Let denote an ndimensional 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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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197403301960
PII:
S 00029947(1974)03301960
Article copyright:
© Copyright 1974
American Mathematical Society
