Website owner: James Miller
An interpretation of the product of two matrices
Suppose that the matrix A carries the vector X1 into the vector Y1, the vector X2 into the vector Y2 , etc. i.e.
AX1 = Y1
AX2 = Y2
......
......
AXn = Yn
Then
A [ X1 X2 .... Xn] = [Y1 Y2 ... Yn]
where [ X1 X2 .... Xn] is a matrix whose columns are X1, X2, .... , Xn and [Y1 Y2 ... Yn] is a matrix whose columns are Y1,Y2, ... ,Yn .
The above stated fact provides us with a way of viewing the product of one matrix by another. In the product
AB = C
the columns of B can be viewed as a set of vectors which are carried by the matrix A into the columns of C (i.e. the matrix A is viewed as an operator that operates on the columns of B, viewed as vectors, and carries them into the columns of C, also viewed as vectors).
In the same way B can be viewed as the operator matrix and the rows of A viewed as vectors which are carried into the rows of C (viewed as vectors).