Condition number and matrices

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.