Extended Index Matrices - Index Matrices: Towards an Augmented Matrix Calculus - page 53

| (

a k f , l g ) =

.

A A

Let for i = 1 , 2 ,..., m :

a k i , f , l i , g b i p i , r , q i , s }] , =[ P i , Q i , {

where for every i , j (1 ≤ i < j ≤ m ):

P i ∩ P j = Q i ∩ Q j =∅ ,

P i ∩ K = Q i ∩ L =∅ .

Then, for k 1 , f , k 2 , f ,..., k m , f ∈ K and l 1 , g , l 2 , g ,..., l m , g ∈ L :

a k 1 , f , l 1 , g , a k 2 , f , l 2 , g ,..., a k m , f , l m , g ) A | (

a k 1 , f , l 1 , g )) | ( a k 2 , f , l 2 , g )) . . .) | ( a k m , f , l m , g ).

= (. . . (( | (

A

Theorem 6 Let the IM A be given and let fori = 1 , 2: k 1 , f = k 2 , f andl 1 , g = l 2 , g

and

a k i , f , l i , g b i p i , r , q i , s }] , =[ P i , Q i , {

where

P 1 ∩ P 2 = Q 1 ∩ Q 2 =∅ ,

P i ∩ K = Q i ∩ L =∅ .

Then ,

a k 1 , f , l 1 , g , a k 2 , f , l 2 , g ) = a k 2 , f , l 2 , g , a k 1 , f , l 1 , g ).

A | ( A | (

l 2 , g is important
(it was omitted in [16]), because if we have two elements of a given IM, that are IMs
and that belong to one row or column, this will generate problems.
Let us give an example. Let the IM A have the form

As it is mentioned in [14], the condition k 1 , f = k 2 , f and l 1 , g =

l 1 l 2 l 3
k 1 a 1 , 1 a 1 , 2 a 1 , 3 A =

k 2 a 2 , 1 a 2 , 2 a 2 , 3 .