Linear algebra matrix norm

[civic forge]

How to show in matrices that $||AB|| \leq ||A|| ||B||$ for the norm defined as $||A|| = \sum_{i,j=1}^{n} |a_{ij}|$?

I've tried writing $ ||AB|| = \sum_{i,j=1}^{n} \sum_{k=1}^{n} |a_{ik} b_{kj}| \leq \sum_{i,j=1}^{n} \sum_{k=1}^{n} |a_{ik} b_{kj}|$ and then $||A|| ||B|| = \sum_{i,j=1}^{n} \sum_{k,l=1}^{n} |a_{ij} b_{kl}|.$ But I can't go from one to another. Did I do something wrong?

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[weary grove]

Migamuel

[elfin vessel]

try it for a 2x2

write everything out

with alphabets as variables

[vagrant juniper]

I did that 1.5 years ago, I may have the solution somewhere

[civic forge]

Yeah but the multiplication matrix gets too big

[vagrant juniper]

I remember that the AB norm had to be pretty manipulated, because you had to separate the double sum into a product of sums

[elfin vessel]

I assume these are real matrices

[vagrant juniper]

Also the second thing is wrong, the ||A|| ||B|| is a product of sums, not a double sum, btw

It may be expressed as a double sum

But in the first step it is not a double sum

[vagrant juniper]

Si no recuerdo mal, busca en los ejercicios de la sección 7.1 del libro de análisis numérico de Burden y Faires, creo que uno de los ejercicios era justo ese

[vagrant juniper]

Y las soluciones están en Internet

[elfin vessel]

on the the left hand side we only have |a(ik) b(kj)| for k=1,...,n and on the right side we have |a(ik)b(kj)| too for k=1,...,n and much more

pairs of form (i, k) and (k, j) versus pairs of form (i, j) and (k, l) the 2nd clearly contains the first

[gilded rock]

You could say $|b_{k,j}| \le \sum_{l=1}^{n} |b_{l,j}|$ for any $k,j$

[weary grove]

Rotor 🙂

[civic forge]

Yeah sorry I skipped the first step

[civic forge]

Hmm yeah I can see that

[civic forge]

oh yeah this works

[civic forge]

Sí, el ejercicio 7.1.2 y justo no lo tienen

[vagrant juniper]

Vale, ahora no estoy en casa, dentro de un rato llego, creo que lo tengo hecho

[vagrant juniper]

I have found it

[twilit fjord]

@civic forge

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[civic forge]

Gracias!

[vagrant juniper]

Si necesitas ayuda con algo más, a lo mejor puedo ayudar

[civic forge]

De momento voy bien, si necesito algo más te pingeo