27th Conference of the International Linear Algebra Society (ILAS 2026)

Session

Matrix Geometries

MS 04

May 20, 2026, 10:45 AM

404. Special orthogonal, special unitary, and symplectic groups as products of Grassmannians

Xiang Lu (University of Chicago)

5/20/26, 10:45 AM

Matrix Geometries

Minisymposium Talk

We describe a curious structure of the special orthogonal, special unitary, and symplectic groups that has not been observed, namely, they can be expressed as matrix products of their corresponding Grassmannians realized as involution matrices. We will show that $\text{SO}(n)$ is a product of two real Grassmannians, $\text{SU}(n)$ a product of four complex Grassmannians, and $\text{Sp}(2n,...

228. Differential-Geometric View of the Schur--Horn Theorem and Related Convexity Phenomena

Tin-Yau Tam (University of Nevada, Reno)

5/20/26, 11:10 AM

Matrix Geometries

Minisymposium Talk

We present a differential--geometric view of the Schur--Horn theorem and related convexity phenomena. For an $n\times n$ Hermitian matrix $A$ with simple spectrum, the Schur--Horn map
$$ \mu: {\mathrm U}(n) \to \mathbb R^n,\quad \mu(U)=\mathrm{diag}(UA U^{-1}), \qquad U\in {\mathrm U}(n), $$
is shown to be a proper submersion over the relative interior of the Schur--Horn polytope, where...

510. Stochastic approximations with operator means

Pálfia Miklós (Corvinus University of Budapest)

5/20/26, 11:35 AM

Matrix Geometries

Minisymposium Talk

In this talk we investigate zeros of nonlinear operators on the cone of positive definite operators over a Hilbert space. The unique zero of such nonlinear operators define means of positive definite operators. Moreover these nonlinear operators generate strictly exponentially contracting semigroups in some metric defined on positive operators. We survey recent results that establish the...

372. Locally Diffeomorphic Logarithm of Special Orthogonal Matrices

Zhifeng Deng (Xiamen University)

5/21/26, 11:00 AM

Matrix Geometries

Minisymposium Talk

The special orthogonal group $\mathbb{SO}_n$ is a Lie group whose geometry and local structure are encoded by the exponential map on its Lie algebra $\mathbf{Skew}_n$, the set of skew-symmetric matrices. The associated inverse problem---the matrix logarithm---exhibits a highly nontrivial local diffeomorphism structure, and the notion of a nearby logarithm arises naturally as a local inverse of...

66. Stiefel Optimization is NP-hard

Dr Tianyun Tang (University of Chicago)

5/21/26, 11:25 AM

Matrix Geometries

Minisymposium Talk

We show that linearly constrained linear optimization over a Stiefel or Grassmann manifold is NP-hard in general. We show that the same is true for unconstrained quadratic opti- mization over a Stiefel manifold. We will show that unless P = NP, these optimization problems over a Stiefel manifold do not have FPTAS. As an aside we extend our results to flag manifolds. Combined with earlier...

483. On the Identifiability of Transformer Self-Attention

Nathan Henry (University of California, Berkeley)

5/21/26, 11:50 AM

Matrix Geometries

Minisymposium Talk

Multi-head self-attention is a fundamental building block of the transformer architecture in modern machine learning, enabling large language models and much of modern generative AI as we know it. However, some aspects of the self-attention function space remain poorly understood. In particular, its parameterization is non-unique: continuous families of unique weight matrices can induce the...