The solution of the equation XA+AXT=0 and its application to the theory of orbits

describe how to find the general solution of the matrix equation XA+AXT=0, with A∈Cn×n, which allows us to determine the dimension of its solution space. This result has immediate applications in the theory of congruence orbits of matrices in Cn×n, because the set {XA+AXT:X∈Cn×n} is the tangent space at A to the congruence orbit of A. Hence, the codimension of this orbit is precisely the dimension of the solution space of XA+AXT=0. As a consequence, we also determine the generic canonical structure of matrices under the action of congruence. All these results can be directly extended to palindromic pencils A+λAT.