The following linear equations provide a method for systematicly converting any array (x',y') of size 2 into an array (x",y",z") of size 3:
(**)
Then it is possible to convert any array (x,y,z,w) of size 4 into an array (x",y",z") by replacing x',y' in (**) by their equal expressions given in (*):
x" = α11(β11x + β12y + β13z + β14w) + α12(β21x + β22y + β23z + β24w)
=
(α11β11 + α12β21)x +
(α11β12 + α12β22)y +
(α11β13 + α12β23)z +
(α11β14 + α12β24)w
y" = α21(β11x + β12y + β13z + β14w) + α22(β21x + β22y + β23z + β24w)
=
(α21β11 + α22β21)x +
(α21β12 + α22β22)y +
(α21β13 + α22β23)z +
(α21β14 + α22β24)w
z" = α31(β11x + β12y + β13z + β14w) + α32(β21x + β22y + β23z + β24w)
=
(α31β11 + α32β21)x +
(α31β12 + α32β22)y +
(α31β13 + α32β23)z +
(α31β14 + α32β24)w
Therefore,
(***)
x" =
(α11β11 + α12β21)x +
(α11β12 + α12β22)y +
(α11β13 + α12β23)z +
(α11β14 + α12β24)w
y" =
(α21β11 + α22β21)x +
(α21β12 + α22β22)y +
(α21β13 + α22β23)z +
(α21β14 + α22β24)w
z" =
(α31β11 + α32β21)x +
(α31β12 + α32β22)y +
(α31β13 + α32β23)z +
(α31β14 + α32β24)w
The coefficients of equations in (**) and (*) form the following two matrices (3x2) A and (2x4) B respectively:
(#)
A = B = α11 α12 β11 β12 β13 β14 (matrices) α21 α22 β21 β22 β23 β24 α31 α32
The coefficients of equations in (***) form the following 3x4 matrix C:
(###)
C = α11β11 + α12β21 α11β12 + α12β22 α11β13 + α12β23 α11β14 + α12β24 (matrix) α21β11 + α22β21 α21β12 + α22β22 α21β13 + α22β23 α21β14 + α22β24 α31β11 + α32β21 α31β12 + α32β22 α31β13 + α32β23 α31β14 + α32β24
Each element of matrix C is a sum of products. But the inner product of two arrays of the same length is defined as the sum of products of all the corresponding coordinates (entries). The [i,j] entry in matricx C is
This motivates the following rule: