Generalized quadratic matrix manifolds: Riemannian geometry and computations
Abstract
We study the set Gq(P,Q) of real n× p matrices X satisfying X⊤ Q X = P, where the nonsingular matrices P and Q are both symmetric or both skew-symmetric and the set is nonempty. Special choices of P and Q yield the Stiefel, generalized Stiefel, indefinite Stiefel, and symplectic Stiefel manifolds, as well as the quadratic matrix Lie groups. We prove that Gq(P,Q) is a closed embedded submanifold of ℝn × p and that it is compact exactly when P and Q are both symmetric and definite of the same sign. We derive closed-form expressions for its tangent space and, under a canonical-like Riemannian metric, for the normal space, the orthogonal projection, and the Riemannian gradient and Hessian. We extend the Cayley, quasi-geodesic, and polar retractions to Gq(P,Q), with explicit domains, vector transports, and closed-form second-order corrections. These objects are the ingredients of Riemannian first- and second-order methods. In trace minimization tests with symmetric positive-definite, skew-symmetric, and symmetric indefinite pairs (P,Q), the Riemannian conjugate gradient and trust-region methods reach the same objective value with different retractions and keep the iterates on Gq(P,Q) up to rounding error.
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BibTeX
@misc{peng2026generalized,
title = {{Generalized quadratic matrix manifolds: Riemannian geometry and computations}},
author = {Zhen Peng and Bin Gao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28159},
eprint = {2610.28159},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28159},
}
RIS
TY - UNPB
TI - Generalized quadratic matrix manifolds: Riemannian geometry and computations
AU - Zhen Peng
AU - Bin Gao
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28159
UR - https://arxiv.org/abs/2610.28159
AB - We study the set Gq(P,Q) of real n× p matrices X satisfying X^⊤ Q X = P, where the nonsingular matrices P and Q are both symmetric or both skew-symmetric and the set is nonempty. Special choices of P and Q yield the Stiefel, generalized Stiefel, indefinite Stiefel, and symplectic Stiefel manifolds, as well as the quadratic matrix Lie groups. We prove that Gq(P,Q) is a closed embedded submanifold of ℝ^n × p and that it is compact exactly when P and Q are both symmetric and definite of the same sign. We derive closed-form expressions for its tangent space and, under a canonical-like Riemannian metric, for the normal space, the orthogonal projection, and the Riemannian gradient and Hessian. We extend the Cayley, quasi-geodesic, and polar retractions to Gq(P,Q), with explicit domains, vector transports, and closed-form second-order corrections. These objects are the ingredients of Riemannian first- and second-order methods. In trace minimization tests with symmetric positive-definite, skew-symmetric, and symmetric indefinite pairs (P,Q), the Riemannian conjugate gradient and trust-region methods reach the same objective value with different retractions and keep the iterates on Gq(P,Q) up to rounding error.
ER -