Pseudo-orthogonal matrix?

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In summary, the matrix is composed of only ones and zeros, between any two rows there is one and only one common column with a one, and the matrix can be diagonalized. However, it appears that generating matrices up to order 7 is a difficult task.
  • #1
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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  • #2
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!).
 
  • #3
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.
 
  • #5
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.
 

1. What is a pseudo-orthogonal matrix?

A pseudo-orthogonal matrix is a square matrix with real entries that satisfies the condition ATA = I, where AT is the transpose of A and I is the identity matrix. Unlike an orthogonal matrix, a pseudo-orthogonal matrix may not have a determinant of 1.

2. How is a pseudo-orthogonal matrix different from an orthogonal matrix?

An orthogonal matrix is a square matrix with real entries that satisfies the condition ATA = I and has a determinant of 1. A pseudo-orthogonal matrix, on the other hand, may not have a determinant of 1. Additionally, the columns of an orthogonal matrix are orthogonal (perpendicular) to each other, while the columns of a pseudo-orthogonal matrix may not be.

3. What are some applications of pseudo-orthogonal matrices?

Pseudo-orthogonal matrices have applications in areas such as signal processing, image processing, and data compression. They are also used in the field of quantum mechanics to represent quantum gates.

4. What is the relationship between pseudo-orthogonal matrices and unitary matrices?

A pseudo-orthogonal matrix is a generalization of a unitary matrix. While a unitary matrix satisfies the condition A*A = I, where A* is the conjugate transpose of A, a pseudo-orthogonal matrix satisfies the condition ATA = I. Therefore, all unitary matrices are also pseudo-orthogonal matrices, but not all pseudo-orthogonal matrices are unitary.

5. Can a pseudo-orthogonal matrix be diagonalized?

Yes, a pseudo-orthogonal matrix can be diagonalized. This means that it can be expressed as a product of three matrices: A = PDPT, where P is an orthogonal matrix and D is a diagonal matrix. This allows for easier computation and manipulation of the matrix.

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