# A Spectral Generalization of Von Neumann Minimax Theorem

## TL;DR

This paper extends the classical Von Neumann minimax theorem to a spectral setting involving symmetric matrices and the spectraplex, establishing a new minimax equality in this broader context.

## Contribution

It introduces a spectral generalization of the minimax theorem applicable to symmetric matrices and the spectraplex, expanding the theoretical framework of minimax results.

## Key findings

- Spectral minimax property holds for symmetric matrices.
- Reduces to classic minimax for diagonal matrices.
- Establishes a new minimax equality in spectral setting.

## Abstract

Given $n \times n$ real symmetric matrices $A_1, \dots, A_m$, the following {\it spectral minimax} property holds: $$\min_{X \in \mathbf{\Delta}_n} \max_{y \in S_m} \sum_{i=1}^m y_iA_i \bullet X=\max_{y \in S_m} \min_{X \in \mathbf{\Delta}_n} \sum_{i=1}^m y_iA_i \bullet X,$$ where $S_m$ is the simplex and $\mathbf{\Delta}_n$ the spectraplex. For diagonal $A_i$'s this reduces to the classic minimax.