Unimodular matrices with additional restrictions

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I have adapted an idea I have found in Stackexchange. It is based on the following facts:

• Let $J_k$ be the identity matrix of order $n$ with the null elements in the $k$-th row replaced by random integers. Then $J_k$ is unimodular, i.e. $\lvert J_k\rvert=1$.

• If $A$ is an unimodular symmetric matrix, so it is $J_k^T A J_k$.

Thus, one can start with $A=I_n$, choose a row $k$ at random, construct $J_k$, compute $J_k^T A J_k$ and replace $A$ by that matrix. This process can be repeated as many times as wanted. The result is an unimodular symmetric matrix. The following code implements this idea:

def unimod_sym_matrix(n, niter=1, min_int=-1, max_int=1):
    A = identity_matrix(n)
    rows = random_vector(niter, x=0, y=n)
    for k in rows:
        J = identity_matrix(n)
        J[k,:] = random_vector(n, x=min_int, y=max_int+1)
        J[k,k] = 1
        A = J.transpose()*A*J
    return A

Observe that, to get $J_k$ , random integers are taken from min_int to max_int, both included. Likewise, niter is the number of matrices $J_k$ that are constructed.

For example, the following code

set_random_seed(1000)
A = unimod_sym_matrix(6,3); show(A)
A = unimod_sym_matrix(6); show(A)

yields

$\begin{pmatrix} 2 & -2 & -1 & 0 & 1 & 0 \\ -2 & 9 & 4 & 1 & -3 & 3 \\ -1 & 4 & 2 & 0 & -1 & 2 \\ 0 & 1 & 0 & 2 & -1 & -1 \\ 1 & -3 & -1 & -1 & 2 & 1 \\ 0 & 3 & 2 & -1 & 1 & 6 \end{pmatrix}$

$\begin{pmatrix} 2 & 0 & 1 & -1 & -1 & -1 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 2 & -1 & -1 & -1 \\ -1 & 0 & -1 & 1 & 1 & 1 \\ -1 & 0 & -1 & 1 & 2 & 1 \\ -1 & 0 & -1 & 1 & 1 & 2 \end{pmatrix}$

I have included set_random_seed just for the sake of reproducibility of the above output.

Since you want matrix entries between $-2$ and $2$, you had better use the default values for niter, min_int and max_int.