"how to test if two matrices are similar by a signed permutation matrix" code sage

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I gave a detailed answer to a fuzzy question by Anonymous in the link below:

https://ask.sagemath.org/question/39515/how-to-find-the-permutation-matrix-associated-with-the-similarity-transformation-in-the-following-code/

(But compared to this question, the 39515 question qualifies as a detailed exposition.)

The steps would be:

• check the characteristic polynoials, if there is no match, we can stop here.
• build the eigenvalues, sort them somehow, if there is no match, stop here.
• build the jordan decomposition and match the eigenvectors, allowing rescaling on the one side, and permutations of rows on the other side. A rough match can be done by first rescaling in the two Jordan base change matrices, so that the maximal absolute value occurs for the entry $1$ in each column.

If you remove "signed" from your question this is equivalent to the isomorphism problem of vertex and edge labeled directed graphs. To do that in Sage let us define the following auxilliary function that turns a matrix into a (edge labeled) directed graph

def mat_to_dig(A):
    n = A.nrows()
    G = DiGraph(loops=False, multiedges=False)
    for i in range(n):
        for j in range(n):
            if A[i,j]:
                G.add_edge(i, j, A[i,j])
    return G

Now, (assuming that I care about matrix with 0 on the diagonal)

sage: A = matrix(4, [0,1,2,3,4,0,5,6,7,8,0,9,10,11,12,0])
sage: A
[ 0   1   2  3]
[ 4   0   5  6]
[ 7   8   0  9]
[10 11 12  0]
sage: p = Permutation('(1,3)(2,4)')
sage: B = copy(A)
sage: B.permute_rows_and_columns(p,p)
sage: B
[ 0   9   7  8]
[12  0 10 11]
[ 2   3   0  1]
[ 5   6   4  0]
sage: G = mat_to_dig(A)
sage: H = mat_to_dig(B)
sage: G.is_isomorphic(H, edge_labels=True, certificate=True)
(True, {0: 2, 1: 3, 2: 0, 3: 1})

The dictionary {0: 2, 1: 3, 2: 0, 3: 1} gives you the permutation to be performed namely 0↦2, 1↦3, 2↦0, 3↦1.

Now, if you want to consider the signed version what you can do is to replace your graph by a "signed graph" in which the vertices are doubled.

def mat_to_signed_dig(A):
    n = A.nrows()
    G = DiGraph(loops=False, multiedges=False)
    for i in range(n):
        for j in range(n):
            if A[i,j]:
                G.add_edge(i, j, A[i,j])
                G.add_edge(n+i, n+j, A[i,j])
                G.add_edge(i, n+j, -A[i,j])
                G.add_edge(n+i, j, -A[i,j])
    return G

Then

sage: A = matrix(4, [0,1,2,3,4,0,5,6,7,8,0,9,10,11,12,0])
sage: p = matrix(Permutation('(1,3)(2,4)')) * diagonal_matrix([1,-1,-1,1])
sage: B = p * A * ~p
sage: B
[  0  -9  -7   8]
[-12   0  10 -11]
[ -2   3   0  -1]
[  5  -6  -4   0]
sage: G = mat_to_signed_dig(A)
sage: H = mat_to_signed_dig(B)
sage: G.is_isomorphic(H, edge_labels=True, certificate=True)
(True, {0: 6, 1: 3, 2: 0, 3: 5, 4: 2, 5: 7, 6: 4, 7: 1})

The dictionary now corresponds to a signed permutation where i+4 should be thought as -i. So it should be interpreted as the signed permutation +0↦-2, +1↦+3, +2↦+0, +3↦-1, -0↦+2, -1↦-3, -2↦-0, -3->+1.