- #1
relinquished™
- 79
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Hi guys,
I know this may sound so "newbieish", but I really need some clarification. While resaerching over the net I came across a proof on a derivation of the Matrix p-norms. While reading, I stumbled upon this part of the proof:
[tex]
\| Ax \|_1 \leq \sum^n_{i=1} \left| \sum^n_{k=1} a_{ik}x_k \right| = \left( \sum^n_{k=1} |x_k| \right) \left( \sum^n_{i=1} |a_{ik}| \right)
\quad (1)
[/tex]
I am confused when I came upon this. I know that
[tex]
\left( \sum_j a_j \right) \left(\sum_k b_k\right) = \sum_j \sum_k a_j b_k \quad (2)
[/tex]
But in (1), the [tex]b_k[/tex] term is given by [tex]a_{ik}[/tex]. Is it possible to "split" the two summation symbols into a product of two sums even though it's clear that [tex]a_{ik}[/tex] is dependent on both i and k?
All help is appreciated
Thanks,
Reli~
I know this may sound so "newbieish", but I really need some clarification. While resaerching over the net I came across a proof on a derivation of the Matrix p-norms. While reading, I stumbled upon this part of the proof:
[tex]
\| Ax \|_1 \leq \sum^n_{i=1} \left| \sum^n_{k=1} a_{ik}x_k \right| = \left( \sum^n_{k=1} |x_k| \right) \left( \sum^n_{i=1} |a_{ik}| \right)
\quad (1)
[/tex]
I am confused when I came upon this. I know that
[tex]
\left( \sum_j a_j \right) \left(\sum_k b_k\right) = \sum_j \sum_k a_j b_k \quad (2)
[/tex]
But in (1), the [tex]b_k[/tex] term is given by [tex]a_{ik}[/tex]. Is it possible to "split" the two summation symbols into a product of two sums even though it's clear that [tex]a_{ik}[/tex] is dependent on both i and k?
All help is appreciated
Thanks,
Reli~
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