Temporal IFIMs - Index Matrices: Towards an Augmented Matrix Calculus - page 69

As above, herewe discuss the operations, relations and operators over the extended
type of TIFIMs.

4.1 Operations Over ETIFIMs

For the ETIFIMs

A ∗ (T ) =[ K ∗ (T ), L ∗ (T ), { μ k i , l j ,τ , ν k i , l j ,τ }] ,

B ∗ (T ) =[ P ∗ (T ), Q ∗ (T ), { ρ p r , q s ,τ , σ p r , q s ,τ }] ,

and for ( ◦ , ∗ ) ∈{ ( max , min ), ( min , max ) } , operations are the following.

Addition

A ∗ (T ) ⊕ ( ◦ , ∗ ) B ∗ (T ) =[ T ∗ (T ), V ∗ (T ), { ϕ t u ,v w ,τ , ψ t u ,v w ,τ }] ,

where

T ∗ (T ) = K ∗ (T ) ∪ P ∗ (T ) ={

t
u

t
u

t u , α

,τ , β ,τ |

t u ∈ K ∪ P & τ ∈ T } ,

V ∗ (T ) = L ∗ (T ) ∪ Q ∗ (T ) ={ v w , α v w,τ , β w,τ | v w ∈ L ∪ Q & τ ∈ T } ,

⎧
⎨

k
i

α

,τ ,

if t u ∈ K − P

p
r

t
u

α

,τ =

α

,τ ,

if t u ∈ P − K ,

⎩

p
r ,τ ),

k
i

max (α

,τ , α

if t u ∈ K ∩ P