# Matrix P-norms are NP-hard to approximate if p \neq 1,2,\infty

## TL;DR

This paper proves that approximating matrix p-norms for most p values is NP-hard, indicating significant computational difficulty unless P equals NP, with implications for algorithms in matrix analysis.

## Contribution

It establishes NP-hardness of approximating matrix p-norms for all rational p except 1, 2, and infinity, extending the understanding of computational complexity in matrix norms.

## Key findings

- NP-hardness of approximating matrix p-norms for p ≠ 1, 2, ∞
- No polynomial-time algorithms for approximating mixed norms within fixed relative precision
- Results hold unless P = NP, indicating fundamental computational limits

## Abstract

We show that for any rational p \in [1,\infty) except p = 1, 2, unless P = NP, there is no polynomial-time algorithm for approximating the matrix p-norm to arbitrary relative precision. We also show that for any rational p\in [1,\infty) including p = 1, 2, unless P = NP, there is no polynomial-time algorithm approximates the \infty, p mixed norm to some fixed relative precision.