A Riemann matrix is a symmetric matrix whose imaginary part is strictly positive definite. In the context of algebraic curves, such a matrix is obtained as a normalized periodmatrix of the algebraic curve.

A Siegel transformation is a transformation from the canonical basis of the homology of a Riemann surface to a new canonical basis of the homology on the Riemann surface such that:

The real part of the new Riemann matrix has entries that are less than or equal to .

The length of the shortest element of L has a lower bound of ,

 : {,  an integer vector} has an upper bound depending only on R and g (=dimension of B) (thus not on B).

The Siegel(B) command returns a list  where  is the new Riemann matrix, and  is the symplectic transformation matrix on the canonical basis of the homology such that the Riemann matrix in the new basis is . If B is a  by  matrix, then  is a  by  matrix. If , where , and  are  by  matrices, the new Riemann matrix is .

Deconinck, B., and van Hoeij, M. "Computing Riemann Matrices of Algebraic Curves." Physica D Vol 152-153, (2001): 28-46.

Siegel, C. L. Topics in Complex Function Theory. Vol. 3. Now York: Wiley, 1973.