有限域中集合{(P,Q)∈GF_s(q)×GF_n(q)|PAQ=B}的基数

The Size of {(P,Q)∈GF_s(q)×GF_n(q)|PAQ=B} in Finite Field

设 A、B是有限域 Fq上两个 s× n级矩阵 ,并且它们的秩都是 r,则存在 Fq上 s级可逆矩阵 P,n级可逆矩阵 Q,使得 PAQ=B.本文讨论有多少对这样的 (P,Q) ,使得 PAQ=B.

Let A?B be two matrices ordered s×n over finite field F , and their ranks are r , then there exists invertible matrice P ordered s and Q ordered n satisfying PAQ=B . This paper try to find the number of (P,Q) that satisfies PAQ=B .