Making matrices better: Geometry and topology of polar and singular value decomposition
Dennis DeTurck
Amora Elsaify
Herman Gluck
Benjamin Grossmann
Joseph Hoisington
Anusha M.Krishnan
Jianru Zhang
2017
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You don't want to view matrices as 9 entries in an array. So you learn the inner product, rank, and the determinant. Then how do matrices look?
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The two components of O(3) appear as real projective 3-spaces in the 8-sphere, each the core of a open neighborhood of nonsingular matrices, whose cross-sectional fibres are triangular 5-dimensional cells lying on great5-spheres. The common boundary of these two neighborhoods is the 7-dimensional algebraic variety V⁷ of singular matrices.
This variety fails to be a submanifold precisely along the 4-manifold M⁴ of matrices of rank 1. The complement V⁷−M⁴, consisting of matrices of rank 2, is a large tubular neighborhood of a core 5-manifold M⁵ consisting of the “best matrices of rank 2”, namely those which are orthogonal on a 2-plane through the origin and zero on its orthogonal complement.
V⁷ is filled by geodesics, each an eighth of a great circle on the 8-sphere, which run between points of M⁵ and M⁴ with no overlap along their interiors. A circle’s worth of these geodesics originate from each point of M⁵, leaving it orthogonally, and a 2-torus’s worth of these geodesics arrive at each point of M⁴, also orthogonally.
