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Matrix Anal. Appl."],"published-print":{"date-parts":[[2025,12,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>We express parallel transport for several common matrix Lie groups with a family of pseudo-Riemannian metrics in terms of the matrix exponential and exponential actions. The metrics are constructed from a deformation of a bi-invariant metric and are naturally reductive. There is a similar picture for homogeneous spaces when taking quotients satisfying a general condition. In particular, for a Stiefel manifold of orthogonal matrices of size [Formula: see text], we give an expression for parallel transport along a geodesic from time zero to [Formula: see text] that could be computed with time complexity of [Formula: see text] for small [Formula: see text] and of [Formula: see text] for large [Formula: see text], contributing a step in a long-standing open problem in matrix manifolds. A similar result holds for flag manifolds with the canonical metric. 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