### An algorithm that carries a square matrix into its transpose by an involutory congruence transformation

Dragomir Z. Djokovic, F. Szechtman, K. Zhao
2003 The Electronic Journal of Linear Algebra
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 ∈ GLn(F ). Moreover, X can be chosen so that X 2 = In, where In 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 is obtained. As a by-product another interesting result is also established. Let G be a semisimple complex
more » ... emisimple complex Lie group with Lie algebra g. Let g = g 0 ⊕ 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. http://math.technion.ac.il/iic/ela where A 21 = I d and the diagonal blocks are square. By subtracting suitable linear combinations of the first d columns from the other columns (using ECT's), we may Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 10, pp. 320-340, December 2003 http://math.technion.ac.il/iic/ela In order to help the reader visualize the shape of the matrix A at this point, we give an example. We take m = 6 and k = 3. Then A has the form: