Index Matrices with Function-Type of Elements - Index Matrices: Towards an Augmented Matrix Calculus - page 81

5.2 Standard Operations Over IMFEs

The forms of these operations also dependent on the forms of IMFE-elements. The
definitions of four of these operations coincide with the operations over IM from
Sect. 1.2 , respectively. In two of the definitions, there are small differences and by
this reason we give them separately.
Let the IMFEs A

=[ K , L , { f k i , l j }] , B =[ P , Q , { g p r , q s }] be given. Then

Addition :

A ⊕ ( ◦ ) B =[ K ∪ P , L ∪ Q , { h t u ,v w }] ,

where

⎧
⎨

f k i , l j , if t u = k i ∈ K and v w = l j ∈ L − Q

or t u = k i ∈ K − P and v w = l j ∈ L ;

g p r , q s , if t u = p r ∈ P and v w = q s ∈ Q − L

or t u = p r ∈ P − K and v w = q s ∈ Q ;

h t u ,v w = ,

⎩

f k i , l j ◦ g p r , q s , if t u =

= ∈ ∩

k i p r K P

and v w = l j = q s ∈ L ∩ Q ;

⊥ , otherwise

where here and below, symbol “ ⊥ ”denotes the lack of operation in the respective

place and' ◦∈{+ , × , max , min ,... } .

Termwise multiplication

A ⊗ ( ◦ ) B =[ K ∩ P , L ∩ Q , { h t u ,v w }] ,

where

h t u ,v w = f k i , l j ◦ g p r , q s ,

for t u =

= ∈ ∩ v w = =

q s ∈

∩ .

k i p r K P and l j L Q

Multiplication

A ( ◦ , ∗ ) B =[ K ∪ ( P − L ), Q ∪ ( L − P ), { c t u ,v w }] ,

where

⎧
⎨

f k i , l j , if t u = k i ∈ K

and v w = l j ∈ L − P − Q

g p r , q s , if t u = p r ∈ P − L − K

and v w = q s ∈ Q

h t u ,v w = ,

⎩

l j = p r ∈ L ∩ P (

◦ f k i , l j ∗ g p r , q s ), if t u = k i ∈ K and v w = q s ∈ Q

⊥ , otherwise