What Is a Pseudo-Orthogonal Matrix?

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Discussion Overview

The discussion revolves around the concept of a "pseudo-orthogonal matrix," which is characterized by specific properties related to its structure and composition. Participants explore its definition, potential applications, and connections to existing mathematical concepts, particularly in the context of combinatorial designs and error-correcting codes.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant describes a pseudo-orthogonal matrix as a binary matrix with equal numbers of ones in each row and column, and with a unique common column of ones between any two rows.
  • Another participant suggests that error correcting code matrices might share properties with the described matrix, although this is presented as a speculative connection.
  • A participant mentions the inspiration for the matrix comes from the card game "Spot it!" which features a unique matching symbol between any two cards, indicating a practical application.
  • One suggestion is made to consider eigenvalues and eigenvectors to explore higher-dimensional matrices related to the topic.
  • A participant reports success in generating pseudo-orthogonal matrices up to order 6 but encounters difficulties with order 7, indicating a potential computational challenge.
  • Reference is made to mutually orthogonal Latin squares as a related concept that may provide further insights into the properties of these matrices.

Areas of Agreement / Disagreement

Participants express varying levels of familiarity with the concept, and while some connections to existing mathematical frameworks are suggested, there is no consensus on the terminology or established properties of the pseudo-orthogonal matrix. The discussion remains exploratory and unresolved.

Contextual Notes

Participants note challenges in generating higher-order matrices and reference existing mathematical literature, indicating a reliance on definitions and properties that may not be fully established in the context of the discussion.

orthogonal
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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.
 
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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!).
 
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.
 
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.
 

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