Given the following information, on $||Ax^{(i)}||$ and $||x^{(i)}||$ for vectors $x^{(1)},x^{(2)},x^{(3)},x^{(4)}$ where A is a nonsingular $n\times n$ matrix.
\begin{array} - & ||x^{(i)}|| & ||Ax^{(i)}|| \\ i=1 & 1 & 100 \\ i=2 & 100 & 1 \\ i=3 & 10^3 & 10^4 \\ i=4 & 10^{-3} & 10^2 \\ \end{array}
I know by definition that $||A||=\max\limits_{x \neq 0} \frac{||Ax||}{||x||}=\max\limits_{||x||=1}||Ax||$ and $||A^{-1}||=\min\limits_{||x||=1} ||Ax||$. Now from the table we can see that $\max\limits_{||x||=1}||Ax||$ is when $i=1$ since here ||x||=1, however, by definition $\max\limits_{x \neq 0} \frac{||Ax||}{||x||}$ is when $i=4$ such that $\max\limits_{x \neq 0} \frac{||Ax||}{||x||}=10^5$. Which is right?
Given the definition, $||A||=\max\limits_{x \neq 0} \frac{||Ax||}{||x||}=\max\limits_{||x||=1}||Ax||$. We can say that $\max\limits_{x \neq 0} \frac{||Ax||}{||x||}$ is not necessarily $||x^{(1)}||$. $||x^{(1)}||$ just happens to have a norm of 1. However this doesn't mean that just because $||x^{(1)}||=1$, then $\max\limits_{||x||=1}||Ax||=100$. Thus, following the defintion and the information given, $||A||=\max\limits_{x \neq 0} \frac{||Ax||}{||x||}=\frac{10^2}{10^{-3}}=10^5$ Since this gives us the largest value when calculating for $\max\limits_{x \neq 0} \frac{||Ax||}{||x||}$ for each vector $x^{(i)}$. Similarly, since $||A^{-1}||=(\min\limits_{||x||=1}||Ax||)^{-1}$. $||A^{-1}||=(\frac{1}{100})^{-1}=100$. Finally $\kappa(A)=||A|||A^{-1}||=10^5 \dot{} 100 = 10^7$