Index Matrices: Definitions, Operations, Relations - Index Matrices: Towards an Augmented Matrix Calculus - page 20

p 1 ... p k p k + 1 ... p k + h p k + h + 1 ... p k + h + g

A = a m α 1 ... α k α k + 1 ... α k + h 0 ... 0

a n 0 ... 0 β k + 1 ... β k + h β k + h + 1 ... β k + h + g

that represents simultaneously the two numbers m and n and then the IM

p 1 ... p k p k + 1 ... p k + h p k + h 1 ... p k + h + g

ρ ( + ) ( A , a ) =

a α 1 ... α k α k + 1 + β k + 1 ... α k + h + β k + h β k + h + 1 ... β k + h + g

represents m . n ,theIM

p 1 ... p k p k + 1 ... p k + h p k + h + 1 ... p k + h + g

ρ ( max ) ( A , a ) =

a α 1 ... α k max (α k + 1 ,β k + 1 ) ... max (α k + h ,β k + h )β k + h + 1 ... β k + h + g

represents the least common multiple of m and n , while the IM

p 1 ... p k p k + 1 ... p k + h p k + h + 1 ... p k + h + g

ρ ( min ) ( A , a ) =

a 0 ... 0min (α k + 1 ,β k + 1 ) ... min (α k + h ,β k + h ) 0 ... 0

represents the greatest common divisor of m and n .
Now, we can use operation “reduction”:

p k + 1 ... p k + h
a min (α k + 1 ,β k + 1 ) ... min (α k + h ,β k + h )

ρ ( min ) ( A , a ) ( ⊥ , { p 1 ,..., p k , p k + h + 1 ,..., p k + h + g } ) =

or operation “projection”:

p k + 1 ... p k + h

pr

p k + h } ρ ( min ) (

A , a ) =

(α k + h ,β k + h ) .

{ a } , { p k + h + 1 ,...,

(α k + 1 ,β k + 1 ) ...

a min min

m

(

n ,

) =

By the same way, we can represent, e.g., the result of operation IM a

( , ) − ( + ) ( , ).

IM m a IM n a

1.10 An Example from Graph Theory

Let us have the following oriented graph C

For it, we can construct the ( 0 , 1 ) -IM which is an adjacency matrix of the graph