CHAPTER 03.05: BINARY OPERATIONS:
Multiplying two matrices Theory In
this segment we will talk about how do we multiply 2 matrices. So two
matrices A and B can be multiplied as A times B. Because we can have
multiplication like B times A also, but limiting it to multiplying it to A
times B under what circumstances can we multiply B by A or multiply A by B is
if the number of columns number of columns of A is same as number of rows of
B. So only if the number of columns of A is same as the number of rows of B
only then can you multiply it so let’s see what that means. So
if I have an A matrix and I have m rows and p
columns - so
I gt m rows and p columns
of A. Then if I have the B matrix, only if this one has p rows, is this
matrix multiplication defined and then this can have any number of columns, n
columns. So the number of columns of A has to same as the number of rows of B
and the resulting matrix which we call - I suppose - the C matrix will have
this many rows and this many columns. So that’s how we are at least able to
say whether the matrix multiplication is allowed. Now
how do we find out individual elements of a C matrix is given by a summation
so ijth element of C will be given by the summation
k is go to one to p, which p is the number of
columns of A or the number of rows of B A ik bkj. So you want
to find the ijth element. What you have to do is
you have to take the ikth elements where k ranges
from 1 to p and the kjth element of B k - again
ranges from 1 to p - and do this product. And then you add all of those
products to be able to get C ij. So what that
basically also means is that if we expand this we will get Ai1 B1j plus Ai2
B2j all the way up to Aip Bpj.
So what you are able to see is that all these elements which are taken off
the A matrix let’s suppose are in the ith row. The
column number is changing and then the elements which are taken from B are
all in the jth column and the row number is
changing so that’s how we would find out what the Cij
element will be. This
can be also viewed as that as if we are taking as we just said that hey we
are taking the ith row of A; and then we are
looking at the jth column of B. So we could be
saying that - hey Cij is that we take the a in the row in the ith row
Ai1 all the way up to Aip. And then what we are
going to do is we are going to multiply by column of the B matrix which will
be the jth column so B1j all the way up to Bpj. So we are taking the ith
row of A and taking the jth row of B and we are
going to find this product. So again what we are finding out is this the 1
row and p columns and this one is the p row and 1 column; and this will
result in a 1 by 1 um in a 1 by 1 matrix which is a scalar. And that’s the
one which will be corresponding to the Cijth
element. So you can either view it as this summation right here or you can
look at it as a dot product of these two vectors. And that’s the end of this
segment. |