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

Abstract describe how to find the general solution of the matrix equation XA + AX T = 0 , with A ∈ C n × 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 C n × n , because the set { XA + AX T : X ∈ C n × 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 + AX T = 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 + λ A T .

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