- #1

- 79

- 0

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~

Last edited: