Let \({\mathcal M}(U)\) be the number of a finite set. Let G be a finite field and suppose \(\xi\) is a generating element of G. Let \(G_{m\times n}\) be the set of all \(m\times n\) matrices on G and \(| A|\) be the determinant of a matrix A. The fundamental theorem of the paper is following theorem: Suppose U is a finite additive group and \({\mathcal T}\) is a transformation group on U. All normal forms in U are \(D_ 0=0\), \(D_ 1,...,D_ m\) under a transformation of \({\mathcal T}\) which satisfies conditions such that \[ \begin{alignedat}{3} &(c_ 1)&\\quad& A \sim 0[{\mathcal T}]&quad& \text{ for any \(A\in U\) if and only if \(A=0,\)} \\ &(c_ 2)&\qquad& A \sim-A[{\mathcal T}]&\quad& \text{ for any \(A\in U,\)} \\ &(c_ 3) &\qquad& T(A-B)=T(A)-T(B)&\quad& \text{ for any \(A,B\in U\) and \(T\in {\mathcal T}.\)} \end{alignedat} \] If it is defined that \((A,B)=i\) when \(A-B\sim D_ i[{\mathcal T}]\) for \(A,B\in U\), then elements in U to the defined relation form an m-class associating scheme and parameters of the scheme are \(v=\mu (U)\), \(n_ i=\mu \{A|\) \(A\in U\), \(A\sim D_ i[{\mathcal T}]\}\), \(p^ i_{jk}=\mu \{A|\) \(A\in U\), \(A\sim D_ j[{\mathcal T}]\), \(A-D_ i\sim D_ k[{\mathcal T}]\}\). By the theorem we had constructed schemes to some sets of matrices on G. Suppose \(U=G_{N\times M}\) and the transformation group \[ \begin{split} {\mathcal T}(m)=\{T_{P,Q}| \quad P\in G_{N\times N},\quad Q\in G_{M\times M},\quad | P| \cdot | Q| =\xi^{\alpha m}\text{ for an integral } \alpha,\\ T_{P,Q}(A)=PAQ \text{ for any } A\in U\} \end{split} \] where m is a natural number. When \(m=1\) and \(N\geq M\geq 1\), under transformation of \({\mathcal T}(1)\) all normal forms in U are \[ D_ 0=0,\quad D_ i=\begin{pmatrix} I_ i & 0 \\ 0 & 0 \end{pmatrix} \quad (1\leq i<M),\quad D_ M=\begin{pmatrix} I_ M \\ 0 \end{pmatrix}. \] If it is defined that \((A,B)=i\) when \(A-B\sim D_ k[{\mathcal T}(1)]\) for any \(A,B\in U\), then an M-class associating scheme is established in U. To general m suppose \(N=M\) and under transformation of \({\mathcal T}(m)\) all normal forms in U are \[ D_ 0=0,\quad D_ i=\begin{pmatrix} I_ i & 0 \\ 0 & 0 \end{pmatrix} \quad (1\leq i<M),\quad D_{M+i}=\begin{pmatrix} I_{M-1} & 0 \\ 0 & \xi^ i \end{pmatrix} \quad (0\leq i<m). \] If it is defined that \((A,B)=i\) when \(A-B\in D[{\mathcal T}(m)]\) for any A, \(B\in U\), then an \((M+m-1)\)-class associating scheme is established in U. To determine parameters of these schemes is quite difficult. In this paper, based on careful and skillful study, explicit expressions or recurrence formulas of parameters of these schemes were obtained.
Similar study was made when U is a set of diagonal or anti-symmetric or Hermitian symmetric or Hermitian anti-symmetric matrices on G in the paper.