Geometric integration on manifold of square oblique rotation matrices (Q2784391)

The authors consider the behaviour of matrix differential systems on the manifold of square oblique rotation matrices (an example of which is the oblique Procrustes problem). Although this manifold is equivalent to the set of nonsingular matrices \(Y\) such that \(\text{diag}(Y^T Y)= I_n\), not all quadratic preserving methods are obliqueness preserving. It is shown that Runge-Kutta methods (such as the Gauss methods) which have the property that the stability matrix \(B^T A+ A^TB- B^TB\equiv 0\) have good obliqueness preserving properties, as do projection methods. This is confirmed by some numerical tests.