Extended Index Matrices - Index Matrices: Towards an Augmented Matrix Calculus - page 58

( ≤ ≤ )

Let for s 1 s n :

K p K q L p L q

( ∀ p , q )( 1 ≤ p < q ≤ n )( ∩ = ∩ =∅ )

and

l s , 1 ... l s , j ... l s , n s

k s , 1 a k s , 1 , l s , 1 ... a k s , 1 , l s , j ... a k s , 1 , l s , n s

. . . . . .

... ...

K s L s a k i , l j }] = A s =[ , , { .

k s , i a k s , i , l s , 1 ... a k s , i , l s , j ... a k s , i , l s , n s

. . . . . .

... ...

k s , m a k s , m , l s , 1 ... a k s , m , l s , j ... a k s , m , l s , n s

The first new operation that we call “ composition ” is defined by

s = 1

n

s = 1

n

K s L s { A s | 1 ≤ s ≤ n }=[ , , { c 1 , t 1 , u ,v 1 ,w , c 2 , t 2 , u ,v 2 ,w ,..., c n , t n , u ,v n ,w }] ,

where for r ( 1 ≤ r ≤ n ) :

⎧
⎨ K r and L r a r , k i , l j , if t u =

∈ v w = ∈

k i l j

c r , t u ,v w =

⎩

⊥ , otherwise

Therefore, it is composed of a new EIM on the basis of n EIMs. The new EIM
contains n -dimensional vectors as elements. By this reason, we define function dim ,
giving the dimensionality of the elements of the EIM A , i.e., for the above EIM, the
equality dim

n holds.
The second new operator, that we call “ automatic reduction ” is defined for a
given EIM A by

( A ) =

@ ( A ) =[ P , Q , { b p r , q s }] ,

⊆ , ⊆

where P K Q L are index sets with the following property:

( ∀ k ∈ K − P )( ∀ l ∈ L )( a k i , l j =⊥ ) & ( ∀ k ∈ K )( ∀ l ∈ L − Q )( a k i , l j =⊥ )

& ( ∀ p r = a i ∈ P )( ∀ q s = b j ∈ Q )( b p r , q s = a k i , l j ).

For example, if

de f g
a 12 ⊥ 3

A = ,

b ⊥⊥⊥ ⊥

c 45

⊥⊥