Three Dimensional IMs - Index Matrices: Towards an Augmented Matrix Calculus - page 94

[3,2,1] -transposition

⎛ ⎞ [3 , 2 , 1]

H K

⎝ ⎠

=

L L

K H

a k i , l j , h g }] [ 3 , 2 , 1 ] =[ [ K , L , H , { H , L , K , { a h g , l j , k i }];

, opera-
tions that are analogous of the usual matrix operations of addition and multiplication
are defined, as well as other, specific ones.

For the 3D-IMs A =[ K , L , H , { a k i , l j , h g }] , B =[ P , Q , R , { b p r , q s , e d }]

Addition

A ⊕ ( ◦ ) B =[ K ∪ P , L ∪ Q , 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 and x y = h g ∈ H − R

or t u = k i ∈ K ,v w = l j ∈ L − Q and x y = h g ∈ H

or t u = k i ∈ K − P ,v w = l j ∈ L and x y = h g ∈ H

b p r , q s , e d , if t u = p r ∈ P ,v w = q s ∈ Q and x y = e d ∈ R − H

or t u = p r ∈ P ,v w = q s ∈ Q − L and x y = e d ∈ R

=

or t u = p r ∈ P − K ,v w = q s ∈ Q and x y = e d ∈ R

⎩

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

and x y = h g = e d ∈ H ∩ R

0 , otherwise

Termwise multiplication

⊗ ( ◦ ) =[ ∩ , ∩ , ∩ , {

c t u ,v w , x y }] ,

A B K P L Q H R

where

c t u ,v w , x y = a k i , l j , h g ◦ b p r , q s , e d ,

for t u = k i = p r ∈ K ∩ P ,v w = l j = q s ∈ L ∩ Q and x y = h g = e d ∈ H ∩ R ;