Index Matrices with Function-Type of Elements - Index Matrices: Towards an Augmented Matrix Calculus - page 85

5.4 Operations Over IMFEs and IMs

Let the IM A =[ K , L , { a k i , l j }] , where a k i , l j ∈ R and IMFE F =[ P , Q , { f p r , q s }]

be given. Then, we can define:

(a) A F =[ K ∪ P , L ∪ Q , { h t u ,v w }] , where

⎧
⎨

a k i , l j . f p r , q s , if t u = k i = p r ∈ K ∩ P

and v w = l j = q s ∈ L ∩ Q ;

h t u ,v w = ,

⎩

⊥ , otherwise

1 ;

with elements of F

(b) A F =[ K ∩ P , L ∩ Q , { h t u ,v w }] , where

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

1 ;

for t u = k i = p r ∈ K ∩ P and v w = l j = q s ∈ L ∩ Q with elements of F

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

⎧
⎨

f p r , q s ( a k i , l j ), if t u =

= ∈ ∩

k i p r K P

and v w = l j = q s ∈ L ∩ Q

h t u ,v w =

⎩

⊥ , otherwise

with elements of R ;

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

h t u ,v w = f p r , q s ( a k i , l j ),

for t u = k i = p r ∈ K ∩ P and v w = l j = q s ∈ L ∩ Q with elements of R .

a k i , l j ,..., a k i , l j }] Let the IM A =[ K , L , { , for the natural number n ≥ 2, where

a k i , l j ,..., a k i , l j

n

∈ R and IMFE F =[ P , Q , { f p r , q s }] , where f p r , q s : F → F be

given. Then

(e) F ♦ ⊕ A =[ K ∪ P , L ∪ Q , { h t u ,v w }] , where

⎧
⎨

a k i , l j ,..., a k i , l j ), f p r , q s ( if t u = k i = p r ∈ K ∩ P

h t u ,v w = and v w = l j = q s ∈ L ∩ Q

⎩

⊥ , otherwise

with elements of R ;