Re: Find an invertible integer matrix that satisfies given conditions.

[Charles Greathouse <crgreathouse@gmail.com>]

Playing around  with linear algebra gives, as an example,

[4,-1,1,4; 4, 4, 0, 5; 4, -1, 5, 4; 4, 4, 0, 9]

There are 8 degrees of freedom here so you have lots of choices to make.

I'll let the real PARI experts suggest code here.

On Tue, Apr 11, 2023 at 9:21 PM Hongyi Zhao <hongyi.zhao@gmail.com> wrote:

Hi here,

I am trying to write a script to find a matrix that satisfies certain
conditions. Specifically, I am trying to find an n x n matrix M that
satisfies the following two conditions:

1. The determinant of M is either 1 or -1.
2. M1 * M = M * M2, where M1 and M2 are two pre-defined matrices.

I have written the following code, but it can't solve the problem:

```
/* Define the two matrices */
M1 = [0, 1, 0, 0; -4, 0, 0, 0; 0, 0, 0, 1; 0, 0, -4, 0];
M2 = [0, 1, 4, 0; -4, 0, 0, -4; 0, 0, 0, 1; 0, 0, -4, 0];

/* Set the size of the matrix */
n = 4;

/* Initialize a flag variable to check if a solution has been found */
flag = 0;

/* Loop until a solution is found */
while(flag == 0,

    /* Generate a random integer matrix */
    M = matrix(n,n,i,j,random(-10,10));

    /* Check if the matrix satisfies the conditions */
    if(abs(Mat(det(M))) == 1 && Mat(M1 * M) == Mat(M * M2),

        /* If the conditions are satisfied, print the matrix and exit
the loop */
        print("Found solution:");
        print(M);
        flag = 1;
    );
);
```

In fact, this is a harder problem as described here [1], I'm not sure
whether it's possible to tackle it in PARI/GP? Thank you in advance
for your help.

[1] https://arxiv.org/abs/1811.06190

Best regards,
Zhao
--
Assoc. Prof. Hongsheng Zhao <hongyi.zhao@gmail.com>
Theory and Simulation of Materials
Hebei Vocational University of Technology and Engineering
No. 473, Quannan West Street, Xindu District, Xingtai, Hebei province