# The GSVD: Where are the ellipses?, Matrix Trigonometry, and more

## TL;DR

This paper develops a geometric and theoretical framework for the Generalized Singular Value Decomposition (GSVD), revealing its natural coordinates, applications, and advantages over traditional eigenproblem approaches, with implications across various scientific fields.

## Contribution

It introduces a geometric interpretation of the GSVD, including an ellipse picture and multiaxes, and advocates for its natural application setting over the eigenproblem formulation.

## Key findings

- The GSVD provides a natural coordinate system for the Grassmann manifold.
- The geometric ellipse interpretation clarifies the role of generalized singular vectors.
- Applications include regularization, genome reconstruction, signal processing, and statistical analysis.

## Abstract

This paper provides an advanced mathematical theory of the Generalized Singular Value Decomposition (GSVD) and its applications. We explore the geometry of the GSVD which provides a long sought for ellipse picture which includes a horizontal and a vertical multiaxis. We further propose that the GSVD provides natural coordinates for the Grassmann manifold. This paper proves a theorem showing how the finite generalized singular values do or do not relate to the singular values of $AB^\dagger$. We then turn to the applications arguing that this geometrical theory is natural for understanding existing applications and recognizing opportunities for new applications. In particular the generalized singular vectors play a direct and as natural a mathematical role for certain applications as the singular vectors do for the SVD. In the same way that experts on the SVD often prefer not to cast SVD problems as eigenproblems, we propose that the GSVD, often cast as a generalized eigenproblem, is rather best cast in its natural setting. We illustrate this theoretical approach and the natural multiaxes (with labels from technical domains) in the context of applications where the GSVD arises: Tikhonov regularization (unregularized vs regularization), Genome Reconstruction (humans vs yeast), Signal Processing (signal vs noise), and statistical analysis such as ANOVA and discriminant analysis (between clusters vs within clusters.) With the aid of our ellipse figure, we encourage in the future the labelling of the natural multiaxes in any GSVD problem.