Angles between subspaces and their tangents

TL;DR

This paper introduces a novel method to explicitly construct matrices whose singular values are equal to the tangents of principal angles between subspaces, expanding the classical understanding based on cosines and sines.

Contribution

It provides new constructions for the tangents of principal angles using various basis and projector approaches, with applications in eigenvalue problem analysis.

Findings

Constructed matrices with singular values equal to tangent of principal angles

Applicable to orthonormal and non-orthonormal bases and projectors

Useful for analyzing convergence in subspace iteration methods

Abstract

Principal angles between subspaces (PABS) (also called canonical angles) serve as a classical tool in mathematics, statistics, and applications, e.g., data mining. Traditionally, PABS are introduced via their cosines. The cosines and sines of PABS are commonly defined using the singular value decomposition. We utilize the same idea for the tangents, i.e., explicitly construct matrices, such that their singular values are equal to the tangents of PABS, using several approaches: orthonormal and non-orthonormal bases for subspaces, as well as projectors. Such a construction has applications, e.g., in analysis of convergence of subspace iterations for eigenvalue problems.