# Condition number ¶

Go back

The condition number (Conditionnement) is a measure of the dependence between the vector of parameters $X$ and the solution $b$.

$cond(A) = |||A||| * |||A^{-1}|||$

Properties

• if $A \in Gl_n(\mathbb{R})$ then $cond(A) \ge 1$
• if $A \in Gl_n(\mathbb{R})$ then $cond(\lambda{A}) = cond(A)$
• if $A, B \in Gl_n(\mathbb{R})$ then $cond(AB) \le cond(A) * cond(B)$

If you forgot, $A \in Gl_n(\mathbb{R})$ means that a matrix of reals $\mathbb{R}$ is invertible.

## Notes ¶

$\begin{split} cond_2(A) = \sqrt{\frac{\sigma_n}{\sigma_1}} \\ or \ if \ A \ is \ positive \ definite \\ cond_2(A) = \sqrt{\frac{\lambda_n}{\lambda_1}} \\\end{split}$

Note that

• $\sigma$: the smallest eigenvalue
• $\lambda$: the biggest eigenvalue

$cond_p(A) = |||A||_p |||A^{-1}||_p$