Speaker
Tin-Yau Tam
(University of Nevada, Reno)
Description
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 ${\mathrm U}(n)$ is the unitary group. We obtain a smooth path-lifting property and a global smooth selection along any line segment in the interior, providing a geometric strengthening of the classical majorization theorem and a proof of Westwick's $c$-numerical range theorem. The talk will emphasize the matrix geometry and topology.
Author
Tin-Yau Tam
(University of Nevada, Reno)