Riemannian geometry

Positive-definite matrix

Quantum mechanics

Engineering

Philosophy

Pure mathematics

Riemannian manifold

Rank (graph theory)

Dimension (graph theory)

Eigenvalues and eigenvectors

Economics

Geodesic

Simple (philosophy)

Mechanical engineering

Physics

Mathematics

Epistemology

Mathematical analysis

Operations management

Geometry

Combinatorics

Logarithm

Metric (unit)

Manifold (fluid mechanics)

Quotient

Generalization

Rayleigh quotient

SIAM Journal on Matrix Analysis and Applications

This paper explores the well-known identification of the manifold of rank $p$ positive-semidefinite matrices of size $n$ with the quotient of the set of full-rank $n$-by-$p$ matrices by the orthogonal group in dimension $p$. The induced metric corresponds to the Wasserstein metric between centered degenerate Gaussian distributions and is a generalization of the Bures--Wasserstein metric on the manifold of positive-definite matrices. We compute the Riemannian logarithm and show that the local injectivity radius at any matrix $C$ is the square root of the $p$th largest eigenvalue of $C$. As a result, the global injectivity radius on this manifold is zero. Finally, this paper also contains a detailed description of this geometry, recovering previously known expressions by applying the standard machinery of Riemannian submersions.

Quotient Geometry with Simple Geodesics for the Manifold of Fixed-Rank Positive-Semidefinite Matrices