Given a Seifert surface F of a knot K in S3, take a1, a2, ..., an to be a basis for the first homology group of F. Then the i,j entry of the Seifert matrix is obtained by taking the linking number of ai & aj#, where aj# indicates a copy of the curve aj which has been pushed slightly off F in the positive normal direction.
Though not itself an invariant of knots, the Seifert matrix can be used to compute knot invariants such as signature, Alexander polynomial, and determinant.