# Condition number and matrices

## TL;DR

This paper clarifies the distinction between the condition number of a matrix and the condition number of a problem, providing a general framework and discussing the classic matrix condition number.

## Contribution

It introduces the general concept of condition number, emphasizing its problem-dependent nature, and applies it specifically to real and complex matrices.

## Key findings

- Condition numbers are associated with problems, not just individual matrices.
- The paper reviews known results about the classic matrix condition number.
- It highlights the importance of problem context in understanding condition numbers.

## Abstract

It is well known the concept of the condition number $\kappa(A) = \|A\|\|A^{-1}\|$, where $A$ is a $n \times n$ real or complex matrix and the norm used is the spectral norm. Although it is very common to think in $\kappa(A)$ as "the" condition number of $A$, the truth is that condition numbers are associated to problems, not just instance of problems. Our goal is to clarify this difference. We will introduce the general concept of condition number and apply it to the particular case of real or complex matrices. After this, we will introduce the classic condition number $\kappa(A)$ of a matrix and show some known results.