Generalized Product Spaces

An Outer Product Space

Here is an idea for a new function in mathematics. I looked for it but could not find it existing already, so I thought I should share it.

We can have an inner product space :

Inner product: (1 2 3) multiply (1 2 3) = 1 4 9

and an outer product space:

Output product space: (1 2 3) multiply (1 2 3) =

1 2 3
2 4 6
3 6 9

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The product space is built from a binary operator. Here the binary operator is multiplication, but it can be any binary operator: plus, mod, power, choose, etc.

But product spaces seem so all or nothing: Inner or Outer.

What if we could create a gradual transition between inner and outer product spaces, a generalized product space, a GPS.

Well, I created the following algorithm that computes an inner and outer product in the extreme cases. I am sure there are other generalized product spaces, but this one seemed to be the simplest. Often the simplest function gives the greatest insight. It is written in python for brevity.

def gps():
    max_i = 4
    matrix = []
    for limit_i in range(4):
        plane = []
        for j in range(4):
            row = []
            for i in range(limit_i+1):
                x = (j + i) % max_i
                row.append( (x+1)*(j+1) )
            plane.append( row)
        matrix.append( plane)

    for limit_i in range(4):
        print ("\n", "gps", 4, 4, "mul", limit_i + 1, end="")
        for j in range(4):
            print(" ")
            for i in range( limit_i+1):
                count_i = (i-j) % (limit_i + 1)
                print( matrix[limit_i][j][count_i], end="\t")

gps()

 gps 4 4 mul 1 
1  
4  
9  
16 
 gps 4 4 mul 2 
1 2  
6 4  
9 12  
4 16 
 gps 4 4 mul 3 
1 2 3  
8 4 6  
12 3 9  
16 4 8 
 gps 4 4 mul 4 
1 2 3 4  
2 4 6 8  
3 6 9 12  
4 8 12 16 

We can see the output of the program. It shows a function that takes a binary function and applies it to two vectors, 1 2 3 4 and 1 2 3 4 and gives an outer product and an inner product space as output. But we can also give it a parameter, m, ranging from 1 to n.

In the program, m is called limit_i. If m = 1 we get an inner product space. If m = n then we get an outer product space. If 1 < m < n, then we get a rectangular matrix whose size is between 1 x N and N x N. If this has any use, who knows? It seemed to me to be a hole in the tapestry of mathematics. Enjoy.