how can I generate all possible mutually orthogonal permutative matrices of order 3 in matlab ?

e.g

1  2  3          1  2  3             (1,1)  (2,2)  (3,3)
2  3  1          3  1  2             (2,3)  (3,1)  (1,2)
3  2  1          2  1  3             (3,2)  (2,1)  (1,3)

these are mutually orthogonal because there is no repition in the ordered pairs.

[David Goodmanson]

Hi Noor,

Calling these two matrices A and B, there are 12 possible A matrices (using 1,2,3, in various arrangements) and all of them can be reduced to the one that you have by interchanging rows and columns (six ways to arrange the first row by swapping columns, and after that you can swap the second and third rows, or not). Looking at it the other way, if you start with the A that you have, you can obtain all the others by interchanging rows and columns.

If A and B are orthogonal, then if you interchange rows and columns in the same way for both of them, orthogonality is preserved.

If you consider your A to be the basic one, you should be able to find six orthogonal B's. Then if you consider the A's to be truly different, you can interchange rows and columns of the solution to create the 12 A's, and six B's for each one of the A's.

[Ammy]

RespectedSir,

Thank you for your kind consideration.

There are 120 permutative matrices of order 3, Is there any way to gererate all pairs out of these 120 which are orthognal to each other .

 I need two orthogonal matrices with the condition that the hamming distance between each row should remain constant .
As in case of order 3 I need two orthogonal matrices with fixed hamming distance 3 between each possible pair of rows .

[David Goodmanson]

Hi Noor,

How is a permutative matrix of order 3 defined? I though it might be a matrix where no number from {1,2,3} is repeated in any row or column, but there are only 12 of those.

Answer 1

[Ammy]

 No number from {1,2,3}  is repeated in row , column repetition is allow.

[David Goodmanson]

Then my previous comment does not have much validity. But it seems like there should be 6x6x6 = 216 possible matrices instead of 120.

[Ammy]

Out of these , Is there any way to generate all possible orthogonal matrices in MATLAB

[Ammy]

I need a MATLAB code that generate all possible pair of orthogonal Permutative Matrices.

Is there any code ?

[David Goodmanson]

One more question on the definition. In your example, the final matrix contained all nine possible ordered pairs. Is that the requirement, or can there be some repetition, such as two different rows containing the same ordered pair?

[Ammy]

No, in the final matrix there should be no repetition in the ordered pairs.

[Ammy]

Out of these nine ordered pairs , there is a possibility that the position of ordered pairs is changed such that the ordered pair locate in the position 11( first row ,first column) may be locate in any other position out of nine ,but with the condition that any ordered pair does not repeat in any position. There should be nine different ordered pairs in the final matrix.

[David Goodmanson]

Hi Noor,

Here is one approach. Consider the first matrix A, it has a 1 somewhere in each of the three rows. In matrix B, those same locations must contain 1,2, and 3, otherwise you would be repeating one of the resulting ordered pairs. So let's say you assign those, and there are 6 different ways to do that.

Matrix A has a 2 somewhere in each of the three rows. In matrix B you have to assign 1,2 and 3 to those locations too, except in a way that does not clash with what you just did, i.e. no 1's in the same row of B, etc. You should be able to fill out the rest of B matrix and come up with two solutions for each of the six possibilities in the previous paragraph.

[Ammy]

Thank you But is there any way to do this in MATLAB or R .

[Ammy]

Is there any MATLAB code which genereate all these possibilities?