Index Matrices: Definitions, Operations, Relations - Index Matrices: Towards an Augmented Matrix Calculus - page 19

pq
a 11 .

IM ( m , a ) = IM ( pq , a ) =

Let us have two natural numbers m and n . In the general case, they have the forms

k + h + g

k + h

p α i
i

p β i
i

m = and n = .

i = 1 i = k + 1

Therefore,

k + h + g

k k + h

p α i
i

p α i + β i
i

p β i
i

mn = ( ).( ).( ).

i = 1 i = k + 1 i = i + h 1

Now, we see that

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

IM ( m , a ) ⊕ ( + ) IM ( n , 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

IM ( m . n , a ),

while if m = n . s , then

m
n ,

IM ( s , a ) = IM ( a ) = IM ( m , a ) − ( + ) IM ( n , a ).

On the other hand,

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

IM ( m , a ) ⊕ ( max ) IM ( n , a ) =

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

that is an IM-interpretation of the least common multiple of m and n . The greatest
common divisor of m and n has an IM-interpretation in the form

p k + 1 ... p k + h

IM ( m , a ) ⊗ ( min ) IM ( n , a ) =

(α k + h ,β k + h ) .

a min (α k + 1 ,β k + 1 ) ... min

Finally, the IM-interpretation of m n

is

...

p 1 p 2 p k

m n IM ( , a ) = n . IM ( m , a ) =

α k .

a n α 1 n α 2 ... n

The result of operation IM ( m , a ) ⊕ ( + ) IM ( n , a ) can be obtained in another way.

We can construct the IM