We study the mathematical structure of the solution set (and its tangent space) to the matrix equation G∗JG=J for a given square matrix J. In the language of pure mathematics, this is a Lie group which is the isometry group for a bilinear (or a sesquilinear) form. The tangent space to G:G∗JG=J consists of solutions to the linear matrix equation X∗J+JX=0. We explicitly demonstrate the computation of the solutions to the equation X∗J±JX=0 for real and complex matrices. We provide numerical examples and visualizations.