Angles between subspaces

Angles between subspaces computed in Clifford Algebra

We first review the definition of the angle between subspaces and how it is computed using matrix algebra. Then we introduce the Grassmann and Clifford algebra description of subspaces. The geometric product of two subspaces yields the full relative angular information in an explicit manner. We explain and interpret the result of the geometric product of subspaces gaining thus full practical access to the relative orientation information.

Blade Products and the Angle Bivector of Subspaces

Principal angles are used to define an angle bivector of subspaces, which fully describes their relative inclination. Its exponential is related to the Clifford geometric product of blades, gives rotors connecting subspaces via minimal geodesics in Grassmannians, and decomposes giving Plücker coordinates, projection factors and angles with various subspaces. This leads to new geometric interpretations for this product and its properties, and to formulas relating other blade products (scalar, inner, outer, etc., including those of Grassmann algebra) to angles between subspaces. Contractions are…

Grassmann angles between real or complex subspaces

The Grassmann angle improves upon similar angles between subspaces that measure volume contraction in orthogonal projections. It works in real or complex spaces, with important differences, and is asymmetric, what makes it more efficient when dimensions are distinct. It can be seen as an angle in Grassmann algebra, being related to its products and those of Clifford algebra, and gives the Fubini-Study metric on Grassmannians, an asymmetric metric on the full Grassmannian, and Hausdorff distances between full sub-Grassmannians. We give formulas for computing it in arbitrary bases, and identitie…

Asymmetric Metrics on the Full Grassmannian of Subspaces of Different Dimensions

Metrics on Grassmannians have a wide array of applications: machine learning, wireless communication, computer vision, etc. But the available distances between subspaces of distinct dimensions present problems, and the dimensional asymmetry of the subspaces calls for the use of asymmetric metrics. We extend the Fubini-Study metric as an asymmetric angle with useful properties, and whose relations to products of Grassmann and Clifford geometric algebras make it easy to compute. We also describe related angles that provide extra information, and a method to extend other Grassmannian metrics to a…

On Jordan angles and triangle inequality in Grassmannian

We obtain the following version of Lidskii theorem. Let L, M, N be p-dimensional subspaces in R^n. Let ψ_j be the angles between L and M, let ϕ_j be the angles between M and N, and let θ_j be the angles between L and N. Consider the orbit of the vector ψwith respect to permutations of coordinates and inversions of axises. Let Z be the convex hull of this orbit. Then θis an element of the polyhedron ϕ+ Z. We discuss similar theorems for other symmetric spaces. We obtain formula for geodesic distance for any invariant Finsler metrics on a classical Riemannian symmetric space.

测试子空间元组之间的同构性

Testing isomorphism between tuples of subspaces

本文提出了一种测试子空间元组同构性的多项式时间算法，并提供了Matlab实现，适用于酉群或一般线性群作用下的同构检测。