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

arXiv: https://arxiv.org/abs/2504.11382 · HTML: https://mlsanity.com/paper/2504.11382/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Apr 2025.

## Built on by (most cited first, 2 of 2)

- [Optimization over the intersection of manifolds](https://mlsanity.com/paper/2605.22736/) (2026)
- [Riemannian Structure and Optimization for a Class of Low-Parametric Orthogonal Matrices](https://mlsanity.com/paper/2609.27982/) (2026)

## Abstract (arXiv)

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.