10 Lectures and 42 Open Problems — Matrix version of 6 deviations suffice

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a real matrix $M\in\mathbb R^{n\times n}$, its operator norm is \[\|M\|_{\mathrm{op}}=\sup_{\|v\|_2=1}\|Mv\|_2.\] Let $H_1,\ldots,H_n$ be real symmetric $n\times n$ matrices with $\|H_k\|_{\mathrm{op}}\le1$ for every $k$. A signing is a vector $\varepsilon\in\{-1,1\}^n$.

Statement recorded in the supplied source.

Is there an absolute constant $C>0$, independent of $n$ and of the matrices, such that for every integer $n\ge1$ and every such family one can choose a signing with \[\left\|\sum_{k=1}^n\varepsilon_k H_k\right\|_{\mathrm{op}}\le C\sqrt n?\] [S2]

Short English statement

Can signs always be chosen so that the operator norm of the signed sum of n symmetric n-by-n contractions is at most a universal constant times the square root of n?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2800403 (AMR-027-0403). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Nikhil Bansal, Haotian Jiang and Raghu Meka, Resolving Matrix Spencer Conjecture Up to Poly-logarithmic Rank (2022), arXiv:2208.11286, Section 1, Conjecture 1.2, p. 1. https://arxiv.org/pdf/2208.11286.