Sub-Symplectic Matrix in Real Field

By replacing the transposed matrix to a real sub-transposed matrix in the definitional expression of symplectic matrix,the concept of real sub-symplectic matrix was proposed.By using the properties of sub-transposed matrix,the properties of transposed matrix,sub-transpose matrix,inverse matrix,adjoint and determinant of real sub-transposed matrix were deduced.Relations between real sub-symplectic matrix and some special matrices,such as sub-symmetric matrix,sub-orthogonal matrix and so on,were discussed.The eigenvalue problem of real sub-transposed matrix was investigated.Real sub-transposed matrix group was proposed and other relevant solutions were obtained.