What Is a Pseudo-Orthogonal Matrix?

[orthogonal]

Hey all, I have been playing around with a special type of matrix and am wondering if anyone knows of some literature about it. I have been calling it a pseudo-orthogonal matrix but would like to learn if it has a real name or if we can come up with a better name. The characteristics of the matrix are as follows:
1) The matrix is composed of only ones and zeros
2) Each row and each column have the same number of ones in it. (If there are 3 ones in each row/column then I call a 3rd order matrix)
3) Between any two rows, there is one and only one common column with a one.

Here is an example of what I call a 3rd order pseudo-orthogonal matrix. Let's call him 'M'
1 1 1 0 0 0 0
1 0 0 1 1 0 0
1 0 0 0 0 1 1
0 1 0 1 0 1 0
0 1 0 0 1 0 1
0 0 1 1 0 0 1
0 0 1 0 1 1 0

I call it a pseudo inverse because inv(M) = M/2-1/6 , i.e. with adding and multiplying by constants I can arrive at the inverse of M.

Has anyone played with something like this before? I am hoping to gleen information to help me generate higher order matrices of this type.

 

[chiro]

Hey orthogonal and welcome to the forums.

I've never played with this kind of thing (I haven't had to): do you have a reason for doing this: (pure curiosity or do you have an application in mind)?

I don't know whether this would help but error correcting code matrices (in binary XOR) might share common properties with this (it's just a hunch and its probably wrong anyway, but you never know!).

 

[orthogonal]

I am working on the applications but I found the matrix by analyzing the matching card game Spot it! It is a card game which has 8 symbols per card with one and only one matching symbol between any two cards. If you follow the link above you can play a demo.

 

[chiro]

How about considering the eigenvalues to create a higher dimensional matrix (as well as the eigenvectors)?

http://en.wikipedia.org/wiki/Matrix_diagonalization

http://en.wikipedia.org/wiki/Eigenvalue#Further_properties

 

[orthogonal]

I have some findings to report:

I have written a program which can generate up to order 6 successfully, but when my code attempts to do order 7 it chokes (24 hours + with no solution returned!).

After googling around some more I found a poster presentation which describes the problem using mutually orthogonal Latin squares. It looks like I have some reading to do to catch up on all this higher order geometry stuff.