Hi, i want to show that zi=cij*xj, where cij=aik*bkj, if yi=bij*xj and zi=aij*yj. (i,j,k are subscripts).
I have been struggling with this for a week. Any help is appreciated, thanks in advance.
In matrix form, you want to show that y = Bx and z = Ay implies that z = (AB)x. From the assumption, it follows that z = A(Bx), so what you need to show is associativity of matrix multiplication: A(Bx) = (AB)x.
Let A, B and C be matrices of compatible sizes. Then
(1)
and
(2)
So we get
I am not sure. One can try writing the sums explicitly instead of using ∑. You can also try becoming more comfortable with ∑-notation and see what distributivity and exchanging limits means when written explicitly. Finally, you may search for explanations of the associativity of matrix multiplication; there must be good descriptions. (I am not sure if you realized that this problems boils down to associativity.)
Feel free to post here what you tried or found.