How do you prove this:
The matrix multipication definition I have is:
Given A = (aij) : m x n matrix
B = (bij) : n x p matrix
We define the product C = AB where C has general element cij = n sigma k=1 aikbkj
=ai1bij + ai2b2j + ai3b3 + .. + ainbnj
How do you prove this:
The matrix multipication definition I have is:
Given A = (aij) : m x n matrix
B = (bij) : n x p matrix
We define the product C = AB where C has general element cij = n sigma k=1 aikbkj
=ai1bij + ai2b2j + ai3b3 + .. + ainbnj
This is a sheer exercise in notation.
Do you know about the Kronecker’s delta notation: $\displaystyle \[
\delta _{jk} = \left\{ {\begin{array}{*{20}c} {1,} & {j = k} \\ {0,} & {j \ne k} \\ \end{array} } \right~?$
It is used to define $\displaystyle I_n=(\delta_{jk}):~n\times n$.
We must also assume that $\displaystyle A=(a_{jk}):~n\times n$.
So $\displaystyle A \cdot I_n = \sum\limits_{j = 1}^n {\left( {\sum\limits_{k = 1}^n{a_{jk} \cdot \delta _{kj} } } \right)} $.
Now ask yourself, ‘If $\displaystyle 1\le J\le n~\&~1\le K\le n $ is it true that
$\displaystyle a_{JK} = \sum\limits_{j = 1}^n {\left( {\sum\limits_{k = 1}^n {a_{Jk} \cdot \delta _{kK} } } \right)}~? $
If so then you are done.
Thanks for that well written explanation. I have heard of the kronecker delta notation but not actually come across it yet in my course.
I don't quite understand the final line though. I understand everything up to that, but not quite how you get to the final part of the proof ..