Three Dimensional IMs - Index Matrices: Towards an Augmented Matrix Calculus - page 98

Second, we define

A ( k , l , h ) = (( A ( k , ⊥ , ⊥ ) ) ( ⊥ , l , ⊥ ) ) ( ⊥ , ⊥ , h ) ,

i.e.,

A

) =[

K −{ k } , L −{ l } , H −{ h } , { c t u ,v w , x y }] ,

( k , l , h

where

c t u ,v w , x y = a k i , l j , h g

for t u = k i ∈ K −{ k } , v w = l j ∈ L −{ l } and x y = h g ∈ H −{ h } .

For every 3D-IM A and for every k 1 , k 2 ∈ K , l 1 , l 2 ∈ L , h 1 , h 2 ∈ H ,

( A

) ) ( k 2 , l 2 , h 2 ) = (

A

) ) ( k 1 , l 1 , h 1 ) .

( k 1 , l 1 , h 1 ( k 2 , l 2 , h 2

Third, let P ={ p 1 , p 2 ,..., p s }⊆ K , Q ={ q 1 , q 2 ,..., q t }⊆ L and R =

{ r 1 , r 2 ,..., r u }⊆ H , p ∈ K , l ∈ L , h ∈ H . Now, we define the following four

operations:

A ( P , l , h ) = (. . . (( A ( p 1 , l , h ) ) ( p 2 , l , h ) )...) ( p s , l , h ) ,

A ( k , Q , h ) = (. . . (( A ( k , l 1 , h ) ) ( k , l 2 , h ) )...) ( k , l t , h ) ,

A ( k , q , H ) = (. . . (( A ( k , l , r 1 ) ) ( k , l , r 2 ) )...) ( k , l , r u ) ,

A ( P , Q , H ) = (. . . (( A ( p 1 , Q , H ) ) ( p 2 , Q , H ) )...) ( p s , Q , H )

= (. . . (( A

) ) ( P , q 2 , H ) )...) ( P , q t , H ) = (...((

A

r 1 ) ) ( P , Q , r 2 ) )...) ( P , Q , r u ) .

( P , q 1 , H ( P , Q ,

Obviously,

A

) =

I

∅ , ( K , L , H

A

( ∅ , ∅ , ∅ ) =

A .

6.5 Operation “Projection” Over an IM

Let P ⊆ K , Q ⊆ L , R ⊆ H . Then,

pr P , Q , R A =[ P , Q , R , { b k i , l j , h g }] ,

where

( ∀ k i ∈ P )( ∀ l j ∈ Q )( ∀ h g ∈ R )( b k i , l j , h g = a k i , l j , h g ).