I have a brief question regarding the infinity matrix norm.
The subordinate matrix infinity norm is defined as:
$\|A\|_{\infty} =\max_{1 \leq i \leq n}\sum_{j=1}^{n}|a_{ij}|.$
This is derived from the general definition of a subordinate matrix norm which is defined as:
$\|A\| = \max \left\{\frac{\|Ax\|}{\|x\|} : x \in K^{n}, x \neq 0\right\}.$
I wanted to try this out in an example. So say we define the matrix:
$A = \begin{bmatrix} 1 & 4 & 2 \\ 3 & 1 & 2 \\ 4 & 4 & 3 \end{bmatrix}$
and
$x = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}.$
Now if we use the first definition, it is easy to see that $\|A\|_{\infty} = 11$
But if we use the general definition, we get:
$\|A\|_{\infty} =\max \left\{\frac{\|Ax\|_{\infty}}{\|x\|_{\infty}} : x \in K^{n}, x \neq 0\right\}.$
Now, we have:
$Ax = \begin{bmatrix} 15 \\ 11 \\ 21 \end{bmatrix}.$
Since the infinity vector norm is defined as:
$\|x\|_{\infty} =\max_{1 \leq i \leq n} |x_i|$
it follows that:
$\|Ax\|_{\infty} = 21$
and:
$\|x\|_{\infty} = 3$
But then we have:
$\frac{\|Ax\|_{\infty}}{\|x\|_{\infty}} = \frac{21}3 = 7$
that does not correlate with the fact that we previously found that $\|A\|_{\infty} = 11$.
If anyone can explain to me what is wrong with my reasoning here, I would appreciate it!