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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Rank $ r$ solutions to the matrix equation $ XAX\sp{T}=C,\,A$ alternate, over $ {\rm GF}(2\sp{y})$


Author: Philip G. Buckhiester
Journal: Trans. Amer. Math. Soc. 189 (1974), 201-209
MSC: Primary 15A33
MathSciNet review: 0330196
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Abstract: Let $ {\text{GF}}(q)$ denote a finite field of characteristic two. Let $ {V_n}$ denote an n-dimensional vector space over $ {\text{GF}}(q)$. An $ n \times n$ symmetric matrix A over $ {\text{GF}}(q)$ is said to be an alternate matrix if A has zero diagonal. Let A be an $ n \times n$ alternate matrix over $ {\text{GF}}(q)$ and let C be an $ s \times s$ symmetric matrix over $ {\text{GF}}(q)$. By using Albert's canonical forms for symmetric matrices over fields of characteristic two, the number $ N(A,C,n,s,r)$ of $ s \times n$ matrices X of rank r over $ {\text{GF}}(q)$ such that $ XA{X^T} = C$ is determined.

A symmetric bilinear form on $ {V_n} \times {V_n}$ is said to be alternating if $ f(x,x) = 0$, for each x in $ {V_n}$. Let f be such a bilinear form. A basis $ ({x_1}, \ldots ,{x_\rho },{y_1}, \ldots ,{y_\rho }),n = 2\rho $, for $ {V_n}$ is said to be a symplectic basis for $ {V_n}$ if $ f({x_i},{x_j}) = f({y_i},{y_j}) = 0$ and $ f({x_i},{y_j}) = {\delta _{ij}}$, for each i, $ j = 1,2, \ldots ,\rho $. In determining the number $ N(A,C,n,s,r)$, it is shown that a symplectic basis for any subspace of $ {V_n}$, can be extended to a symplectic basis for $ {V_n}$. Furthermore, the number of ways to make such an extension is determined.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0330196-0
PII: S 0002-9947(1974)0330196-0
Article copyright: © Copyright 1974 American Mathematical Society