ELA, Volume 10, pp. 320-340, December 2003, abstract.

An Algorithm that Carries a Square Matrix Into Its Transpose
By an Involutory Congruence Transformation

by

D.Z. Djokovic, F. Szechtman and K. Zhao

For any matrix X let X' denote its transpose. It is known that 
if A is an n-by-n matrix over a field F, then A and A' are 
congruent over F, i.e., XAX'=A' for some X in GL_n(F). Moreover, 
X can be chosen so that X^2=I_n, where I_n is the identity matrix.
An algorithm is constructed to compute such an X for a given 
matrix A. Consequently, a new and completely elementary proof of 
that result.

As a by-product another interesting result is also established.
Let G be a semisimple complex Lie group with Lie algebra g. Let 
g be the direct sum of g_0 and g_1 be a Z_2-gradation such that
g_1 contains a Cartan subalgebra of g. Then L.V. Antonyan has 
shown that every G-orbit in g meets g_1. It is shown that, in 
the case of the symplectic group, this assertion remains valid 
over an arbitrary field F of characteristic different from 2. 
An analog of that result is proved when the characteristic is 2.