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

Abstract

Let GF ( q ) {\text {GF}}(q) denote a finite field of characteristic two. Let V n {V_n} denote an n-dimensional vector space over GF ( q ) {\text {GF}}(q) . An n × n n \times n symmetric matrix A over GF ( q ) {\text {GF}}(q) is said to be an alternate matrix if A has zero diagonal. Let A be an n × n n \times n alternate matrix over GF ( q ) {\text {GF}}(q) and let C be an s × s s \times s symmetric matrix over GF ( q ) {\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 ) N(A,C,n,s,r) of s × n s \times n matrices X of rank r over GF ( q ) {\text {GF}}(q) such that X A X T = C XA{X^T} = C is determined. A symmetric bilinear form on V n × V n {V_n} \times {V_n} is said to be alternating if f ( x , x ) = 0 f(x,x) = 0 , for each x in V n {V_n} . Let f be such a bilinear form. A basis ( x 1 , … , x ρ , y 1 , … , y ρ ) , n = 2 ρ ({x_1}, \ldots ,{x_\rho },{y_1}, \ldots ,{y_\rho }),n = 2\rho , for V n {V_n} is said to be a symplectic basis for V n {V_n} if f ( x i , x j ) = f ( y i , y j ) = 0 f({x_i},{x_j}) = f({y_i},{y_j}) = 0 and f ( x i , y j ) = δ i j f({x_i},{y_j}) = {\delta _{ij}} , for each i, j = 1 , 2 , … , ρ j = 1,2, \ldots ,\rho . In determining the number N ( A , C , n , s , r ) N(A,C,n,s,r) , it is shown that a symplectic basis for any subspace of V n {V_n} , can be extended to a symplectic basis for V n {V_n} . Furthermore, the number of ways to make such an extension is determined.