//SINGULAR Example 2.6.3 proc diagonalForm(matrix M) { int n=nrows(M); int m=ncols(M); matrix N,K; matrix L[n][m]; while(N!=M) { N=M; M=L; K=transpose(interred(transpose(interred(N)))); M[1..nrows(K),1..ncols(K)]=K; } return(N); } option(redSB); ring R=0,(x),(C,dp); matrix M[2][3]=(x2+1)^2,0, 0, 0, x3-x-1,0; matrix N1[2][2]=1, 1, 2,-2; matrix N2[3][3]=1,2, 3, 4,5, 6, 7,8,-1; M=N1*M*N2; print(M); diagonalForm(M); matrix M0[5][5]=1, 1,0, 0,0, 3,-1,0, 0,0, 0, 0,1, 1,0, 0, 0,3,-1,0, 0, 0,0, 0,2; matrix N[5][5]=1, 1, -1, 1,-1, 2, 2, 1, 1, 0, -1, 2, 2, 1, 1, -2, 1, 1, -1, 0, 1, 2, -2, 1, 1; M=lift(N,freemodule(nrows(N)))*M0*N-x*freemodule(5); print(M); print(diagonalForm(M));