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The tangent cone to the real determinantal variety: various expressions and a proof

Guillaume Olikier, Petar Mlinarić, P. -A. Absil, André Uschmajew

Abstract

The set of real matrices of upper-bounded rank is a real algebraic variety called the real generic determinantal variety. An explicit description of the tangent cone to that variety is given in Theorem 3.2 of Schneider and Uschmajew [SIAM J. Optim., 25 (2015), pp. 622-646]. The present paper shows that the proof therein is incomplete and provides a proof. It also reviews equivalent descriptions of the tangent cone to that variety. Moreover, it shows that the tangent cone and the algebraic tangent cone to that variety coincide, which is not true for all real algebraic varieties.

The tangent cone to the real determinantal variety: various expressions and a proof

Abstract

The set of real matrices of upper-bounded rank is a real algebraic variety called the real generic determinantal variety. An explicit description of the tangent cone to that variety is given in Theorem 3.2 of Schneider and Uschmajew [SIAM J. Optim., 25 (2015), pp. 622-646]. The present paper shows that the proof therein is incomplete and provides a proof. It also reviews equivalent descriptions of the tangent cone to that variety. Moreover, it shows that the tangent cone and the algebraic tangent cone to that variety coincide, which is not true for all real algebraic varieties.

Paper Structure

This paper contains 23 sections, 5 theorems, 41 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Projection and tangent cone in a Euclidean vector space
  3. Matrix notation
  4. The tangent cone to the real generic determinantal variety
  5. Proof of Theorem \ref{['thm:TanConeRealGenericDeterminantalVariety']}
  6. Proof of the first inclusion in (12)
  7. Preliminaries
  8. Main inequality
  9. Proof of the third inclusion in (12)
  10. The gap in Theorem 3.2 in Schneider/Uschmajew 2015
  11. The algebraic tangent cone to the generic determinantal variety
  12. Elements of algebraic and analytic geometry
  13. Affine algebraic sets
  14. Real and complex analytic sets
  15. Tangent cones to real and complex affine algebraic sets
  16. ...and 8 more sections

Key Result

Proposition 2.1

If $S \subseteq \mathcal{E}$ is a closed cone, then ${T}_{S}^{}(0) = S$.

Figures (1)

  • Figure 1: Artist view of the decomposition used in the proofs of sections \ref{['subsec:GeometricProof']} and \ref{['subsec:AlgebraicProof']}. The picture correctly represents the following aspects: $P_{{T}_{\mathbb{R}_{\mkern 0.9mu\underline{\mkern-0.9mur\mkern-0.9mu}\mkern 0.9mu}^{m \times n}}^{}(X)}(X_i-X)$ is in ${T}_{\mathbb{R}_{\mkern 0.9mu\underline{\mkern-0.9mur\mkern-0.9mu}\mkern 0.9mu}^{m \times n}}^{}(X)$, $X_i-L_i$ and $L_i-X-P_{{T}_{\mathbb{R}_{\mkern 0.9mu\underline{\mkern-0.9mur\mkern-0.9mu}\mkern 0.9mu}^{m \times n}}^{}(X)}(X_i-X)$ are in ${N}_{\mathbb{R}_{\mkern 0.9mu\underline{\mkern-0.9mur\mkern-0.9mu}\mkern 0.9mu}^{m \times n}}^{}(X)$, $L_i$ has rank $\mkern 0.9mu\underline{\mkern-0.9mur\mkern-0.9mu}\mkern 0.9mu$, and $\lVert L_i-X-P_{{T}_{\mathbb{R}_{\mkern 0.9mu\underline{\mkern-0.9mur\mkern-0.9mu}\mkern 0.9mu}^{m \times n}}^{}(X)}(X_i-X) \rVert/\lVert P_{{T}_{\mathbb{R}_{\mkern 0.9mu\underline{\mkern-0.9mur\mkern-0.9mu}\mkern 0.9mu}^{m \times n}}^{}(X)}(X_i-X) \rVert \xrightarrow{i \to \infty} 0$ since $R^\mathrm{orth}$ is a retraction. However, the picture is unfaithful in several respects: $X_i$, $L_i$, and $X+P_{{T}_{\mathbb{R}_{\mkern 0.9mu\underline{\mkern-0.9mur\mkern-0.9mu}\mkern 0.9mu}^{m \times n}}^{}(X)}(X_i-X)$ are, in general, not aligned, and the smooth manifold $\mathbb{R}_{\mkern 0.9mu\underline{\mkern-0.9mur\mkern-0.9mu}\mkern 0.9mu}^{m \times n}$ does not look like that regardless of $m$, $n$, and $\mkern 0.9mu\underline{\mkern-0.9mur\mkern-0.9mu}\mkern 0.9mu$.

Theorems & Definitions (9)

  • Proposition 2.1
  • proof
  • Theorem 4.1
  • Theorem 4.2
  • proof
  • Theorem 7.1
  • proof
  • Lemma A.1
  • proof