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Schubert varieties and distances between subspaces of different dimensions. (English) Zbl 1365.14065
Authors’ abstract: We resolve a basic problem on subspace distances that often arises in applications: How can the usual Grassmann distance between equidimensional subspaces be extended to subspaces of different dimensions? We show that a natural solution is given by the distance of a point to a Schubert variety within the Grassmannian. This distance reduces to the Grassmann distance when the subspaces are equidimensional and does not depend on any embedding into a larger ambient space. Furthermore, it has a concrete expression involving principal angles and is efficiently computable in numerically stable ways. Our results are largely independent of the Grassmann distance – if desired, it may be substituted by any other common distances between subspaces. Our approach depends on a concrete algebraic geometric view of the Grassmannian that parallels the differential geometric perspective that is well established in applied and computational mathematics.

MSC:
14M15 Grassmannians, Schubert varieties, flag manifolds
15A18 Eigenvalues, singular values, and eigenvectors
14N20 Configurations and arrangements of linear subspaces
51K99 Distance geometry
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