3. Partition of the set of nth-order matrices with complex coefficients by similarity classes

3.1 General

Whereas r-equivalence yields a finite number of equivalence classes (parametrized by rank), and congruence yields, in the case of real symmetric matrices, an equally finite number of orbits (parametrized by signature), the number of orbits under the left action of GL(n, $\mathbb{K}$ ) is infinite as soon as $\mathbb{K}$ is infinite. This is because the equivalence classes are then parametrized by the scaled matrices or by the vector subspaces of