Index Matrices: Definitions, Operations, Relations - Index Matrices: Towards an Augmented Matrix Calculus - page 2

where

={

k 1 , k 2 ,..., k m }

={

l 1 , l 2 ,..., l n } ,

K and L

and for 1 ≤ i ≤ m , and for 1 ≤ j ≤ n : a k i , l j ∈ R .

, we obtain a particular case of an IM with
elements being real numbers, that we denote by

When set R is changed with set { 0 , 1 }

( 0 , 1 ) -IM.

1.2 Operations Over R -IMs and ( 0 , 1 ) -IMs

For the IMs A , operations that are anal-

ogous to the usual matrix operations of addition and multiplication are defined, as
well as other, specific ones.

=[ K , L , { a k i , l j }] , B =[ P , Q , { b p r , q s }]

Addition

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

where

⎧
⎨

a 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 ;

b 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 ;

c t u ,v w =

⎩

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

and v w = l j = q s ∈ L ∩ Q

0 , otherwise

◦ ◦ =

Of course, here and below, if “ ” is substituted by “+”, then a k i , l j b p r , q s

a k i , l j + b p r , q s ,while,if“ ◦ ” is “max” or min, then a k i , l j ◦ b p r , q s = max ( a k i , l j , b p r , q s )

or a k i , l j ◦ , respectively.

The geometrical interpretation of operatio n

b p r , q s = min ( a k i , l j , b p r , q s )

⊕ ( ◦ ) is

Termwise multiplication

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