Three Dimensional IMs - Index Matrices: Towards an Augmented Matrix Calculus - page 96

−

where “ ” is the set-theoretic difference operation and

c t u ,v w , x y = a k i , l j , h g , for t u = k i ∈ K − P ,v w = l j ∈ L − Q and x y = h g ∈ H − R .

Multiplication with a constant

α · A =[ K , L , H , { α · a k i , l j , h g }] ,

where α is a constant.

Termwise subtraction

A − ( ◦ ) B = A ⊕ ( ◦ ) ( − 1 ) · B .

The “zero”-3D-IM is

I

∅ =[∅ , ∅ , ∅ , {

a k i , l j , x y }] .

6.3 Relations Over 3D-IMs

Let the two 3D-IMs A =[ K , L , H , { a k , l , h }] and B =[ P , Q , R , { b p , q , e }] be given.

We will introduce the following definitions where ⊂ and ⊆ denote the relations

“strong inclusion” and “weak inclusion”.

The strict relation “inclusion about dimension” is

A ⊂ d B iff ((( K ⊂ P ) & ( L ⊂ Q ) & ( H ⊂ R )) ∨ (( K ⊆ P ) & ( L ⊂ Q ) & ( H ⊂ R ))

∨ (( K ⊂ P ) & ( L ⊆ Q ) & ( H ⊂ R )) ∨ (( K ⊂ P ) & ( L ⊂ Q ) & ( H ⊆ R )))

& ( ∀ k ∈ K )( ∀ l ∈ L )( ∀ h ∈ H )( a k , l , h = b k , l , h ).

The non-strict relation “inclusion about dimension” is

A ⊆ d B iff ( K ⊆ P ) & ( L ⊆ Q ) & ( H ⊆ R ) & ( ∀ k ∈ K )( ∀ l ∈ L )

( ∀ h ∈ H )( a k , l , h = b k , l , h ).

The strict relation “inclusion about value” is

A ⊂ v B iff ( K = P ) & ( L = Q ) & ( H = R ) & ( ∀ k ∈ K )( ∀ l ∈ L )( ∀ h ∈ H )

( a k , l , h < b k , l , h ).