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Generalized quadratic matrix manifolds: Riemannian geometry and computations

Zhen Peng, Bin Gao · 2026 · arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.28159 · cited by 0 works (OpenAlex) · open-access PDF

Hindi

हम वास्तविक $ n\times p$ मैट्रिक्स पर वास्तविक $ n\times p$ $ X$ के विशेष विकल्पों को समायोजित करते हैं जो $ X\top Q X = P$ को संतुष्ट करते हैं, जहां एक ही nonsingular मैट्रिक्स $ P$ और $ Q$ दोनों सिमेट्रिक या स्क्वा-सामेट्रिक होते हैं और सेट nonempty है। $ P$ और $ Q$ के विशेष विकल्प Stiefel, generalized Stiefel, indefinite Stiefel, और symplectic Stiefel manifolds, साथ ही चौथाई मैट्रिक्स Lie समूहों को समायोजित करते हैं। हम साबित करते हैं कि $\mathrm{G}q(P,Q)$ $ के एक बंद अंतर्निहित उप-मैट्रिक्स है $\math

Machine translation (Meta M2M100); the English original follows.

Abstract

We study the set $\mathrm{Gq}(P,Q)$ of real $n\times p$ matrices $X$ satisfying $X^\top Q X = P$, 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 $\mathrm{Gq}(P,Q)$ is a closed embedded submanifold of $\mathbb{R}^{n \times p}$ 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 $\mathrm{Gq}(P,Q)$, 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 $\mathrm{Gq}(P,Q)$ up to rounding error.

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Topics: Mathematics, Fundamental theorem of Riemannian geometry, Riemannian geometry, Symmetric matrix, Tangent space, Mathematical analysis

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