Extended Index Matrices - Index Matrices: Towards an Augmented Matrix Calculus - page 56

l 1 ... l g − 1 q 1 ... q u l g + 1 ... l n

k 1 a k 1 , l 1 ... a k 1 , l g − 1 a k 1 , l g ... a k 1 , l g a k 1 , l g + 1 ... a k 1 , l n

. . . . . . . . . .

k f − 1 a k f − 1 , l 1 ... a k f − 1 , l g − 1 a k f − 1 , l g ... a k f − 1 , l g a k f − 1 , l g + 1 ... a k f − 1 , l n

p 1 a k f , l 1 ... a k f , l g − 1 b p 1 , q 1 ... b p 1 , q v a k f , l g + 1 ... a k f , l n

= .

. . . . . . . . . .

p u a k f , l 1 ... a k f , l g − 1 b p u , q 1 ... b p u , q v a k f , l g + 1 ... a k f , l n

... ... ...

k f + 1 a k f + 1 , l 1 a k f + 1 , l g − 1 a k f + 1 , l g a k f + 1 , l g a k f + 1 , l g + 1 a k f + 1 , l n

. . . . . . . . . .

k m a k m , l 1 ... a k m , l g − 1 a k m , l g ... a k m , l g a k m , l g + 1 ... a k m , l n

Now, the following assertion is valid .
Theorem 9 Let A

=[ K , L , { a k i , l j }] be an IM and let a k f , l g =[ P , Q , { b p r , q s }] be

its element. Then

A | ∗ ( a k f , l g )
= ( A [{ k f } , { l g } , { 0 }] ) ⊕ a k f , l g ⊕[ P , L −{ l g } , { c x , l j }] ⊕ [ K −{ k f } , Q , { d k i , y }] ,

where for each t ∈ P and for each l j ∈ L −{ l g } ,

c x , l j = a k f , l j

and for each k i ∈ K −{ k f } and for each y ∈ Q ,

d k i , y = a k i , l g .

| ∗ (

We can give other representations of IMs A | ( a k f , l g ) and A a k f , l g ) , using other

operations defined over IMs .
The following equalities are valid.

Theorem 10 Let A =[ K , L , { a k i , l j }] be an IM and let a k f , l g =[ P , Q , { b p r , q s }] be

its element. Then

A | ( a k f , l g ) = pr K −{ k f } , L −{ l g } A ⊕ a k f , l g ,

| ∗ ( A a k f , l g )

=

pr K −{ k f } , L −{ l g } A ⊕ a k f , l g ⊕[ P , L −{ l g } , { c x , l j }] ⊕ [ K −{ k f } , Q , { d k i , y }] ,

where for each x ∈ P and for each l j ∈ L −{ l g } ,

c x , l j = a k f , l j