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.