Three Dimensional IMs - Index Matrices: Towards an Augmented Matrix Calculus - page 95

Multiplication
This operation is related to the operation “transposition”. There are six different
operations “multiplication”. The first (standard) multiplication is

[ 1 , 2 , 3 ]
( ◦ , ∗ )

A ( ◦ , ∗ ) B = A B =[ K ∪ ( P − L ), Q ∪ ( L − P ), H ∪ R , { c t u ,v w , x y }] ,

where

c t u ,v w , x y

⎧
⎨

a k i , l j , h g ,

if t u = k i ∈ K

& v w = l j ∈ L − P − Q & x y = h g ∈ H

or t u = k i ∈ K − P − Q

& v w = l j ∈ L & x y = h g ∈ H

b p r , q s , e d , if t u =

∈

p r P

v w =

q s ∈ Q & x y = e d ∈ & R

=

or t u = p r ∈ P − L − K

⎩

& v w = q s ∈ Q & x y = e d ∈ R

◦ a k i , l j , h g ∗ b p r , q s , e d , if t u = k i ∈ K & v w = q s ∈ Q

l j = p r ∈ L ∩ P

& x y = h g = e d ∈ H ∩ R

⊥ , otherwise

The geometrical interpretation of this operation is

Let [ x , y , z ] be a permutation of triple [ 1 , 2 , 3 ] . Operation [ x , y , z ] -multiplication

is defined by:

[ x , y , z ]
( ◦ , ∗ )

B [ x , y , z ] .

= ( ◦ , ∗ )

A B A

Structural subtraction

A B =[ K − P , L − Q , H − R , { c t u ,v w , x y }] ,