The number of solutions to the bilinear matrix equation over a finite field (Q5934186)

Let \(F_q\) be a finite field with \(q\) elements, where \(q\) is a prime power. Using generating matrices of subspaces, the paper gives the number of solutions \((X,Y)\) with \(\text{rank} X=k_1\), \(\text{rank} Y=k_2\) to bilinear matrix equation \(XAY'=B\) over \(F_q\), where \(A\) is an \(m\times n\) matrix and \(B\) is a \(t\times u\) matrix. Moreover, the authors have got the number of all solutions \((X,Y)\) to the above bilinear matrix equation.