Papers › The Geometry of Rank Drop in a Class of Face-Splitting Matrix Products
The Geometry of Rank Drop in a Class of Face-Splitting Matrix Products
Erin Connelly, Sameer Agarwal, Alperen Ergur, Rekha R. Thomas
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Given k ≤6 points (xᵢ,yᵢ) ∈ℙ² ×ℙ², we characterize rank deficiency of the k ×9 matrix Zₖ with rows xᵢ^⊤ ⊗yᵢ^⊤ in terms of the geometry of the point configurations {xᵢ} and {yᵢ}. While this question comes from computer vision the answer relies on tools from classical algebraic geometry: For k ≤5, the geometry of the rank-drop locus is characterized by cross-ratios and basic (projective) geometry of point configurations. For the case k=6 the rank-drop locus is captured by the classical theory of cubic surfaces.
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