THE EQUATION XA + AX ∗ = 0 AND THE DIMENSION OF ∗ CONGRUENCE ORBITS ∗

We solve the matrix equation XA+AX� = 0, where A 2 Cn×n is an arbitrary given square matrix, and we compute the dimension of its solution space. This dimension coincides with the codimension of the tangent space of thecongruence orbit of A. Hence, we also obtain the (real) dimension ofcongruence orbits in C n×n . As an application, we determine the generic canonical structure forcongruence in C n×n and also the generic Kronecker canonical form ofpalindromic pencils A + �A � .

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