Matrix multiplication: index / suffix notation issues

[Dixanadu]

Hey everyone,

I'm struggling with the summation notation for matrices and vector operations, multiplication in particular. Please refer to the image below where I've typed it all out in Word, its too cumbersome here and I want my meaning to be clear:
https://imageshack.us/scaled/large/580/indicesquestion1.jpg

 

[Dixanadu]

Ah man that image turned out tiny...If you guys need it to be bigger please let me know!

 

[AlephZero]

Your first summation for $(AB)_{ij}$ is ok.

Your notation for the next one isn't right. If you want to multiply (AB) by C, you have
$$(ABC)_{ij} = \sum_l^L (AB){}_{il}C_{lj}$$

Now plug in the summation for $(AB)_{il}$ and you get
$$(ABC)_{ij} = \sum_l^L (\sum_k^K A_{ik}B_{kl})C_{lj}$$

You got to the right formula in the end, but (as you said) not in a very logical way.

 

[Dixanadu]

Okay, thank you, I still have a few more questions but for the time being, can you tell me if I can swap the sigma symbols without reordering the suffixes?

 

[AlephZero]

It doesn't matter what order you add up the terms. Changing the order is the same as saying a+b+c+d+e+f = a+d+b+e+c+f, or whatever order you like.

 

[Dixanadu]

Alright. Just to make sure that I understand this properly...let's say I want to multiply 4 matrices A B C D together, where the multiplication is possible. Does it go like this:

https://imageshack.us/scaled/large/853/summationquestion.jpg

Hopefully its right T_T and now the image is too big :(

 

[mikeph]

That's right but I would say make sure you get your brackets right. (AB)C_ij is a very confusing notation, when what you mean is (ABC)_ij. ABC is the matrix, and ij indexes an element in ABC, not just C.

Strictly, (AB)C_ij is the matrix AB times a scalar C_ij.

 

[Dixanadu]

aaah alright. Yea the notation is a bit of an issue but thanks a lot for your help! I got some more questions but I think I'll make a new thread cos its a bit different :)