# Can one help me to find Matlab coding to convert the given matrix to a symmetric matrix by rearranging rows?

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Dr.SIMON RAJ F on 16 Jan 2022
Edited: Torsten on 21 Jan 2022
I would like to covert a mtrix of dimension 12 x12 (Higher dimension )to symmetric matrix by rearranging rows of the marix. As I have matrices with big oder rewriting in symmetrical oder is diffucult.
Dr.SIMON RAJ F on 17 Jan 2022
Thank you Sir. This problem has no solution if A is not SYMMETRICABLE by REARRANGING rows or columns of A. I am not looking for an approximate answer. Kindly see the attachment. I came to arrive this problem when I try to draw a graph (Graph theory) using Matlab. But the matlab gave me error message saing that Adjacensy matrix is not symmetric . So I required the solution for this problem. Kindly help if any other matlab codding available to draw big graphs(Adjacency martix with order more than 10x10).
First step is , We must ensure that the every elements in A must present twice in the matrix A . If not, then there no way A can be symmetricable (if there is a word in english)

Torsten on 17 Jan 2022
Edited: Torsten on 21 Jan 2022
Use intlinprog or bintprog to solve
min: sum_{i=1}^{n} sum_{j=1}^{n} (e_ij+) + (e_ij-)
under the constraints
E+ - E- - (P*A - (P*A)') = 0
sum_{i=1}^{n} p_ij = 1 for all 1 <= j <= n
sum_{j=1}^{n} p_ij = 1 for all 1 <= i <= n
E+, E- >= 0
p_ij in {0,1} for all i and j
If A is "symmetrizable", then the objective will give minimum value 0 and B is equal to P*A.
P is a permutation matrix that "reorders" the rows of A.
Torsten on 21 Jan 2022
A = [13 0 9 -5 4 1 11 -18 16 2 -9 -11 19 -1 -4 -2 -15 4 3 1;
-2 -4 -12 -2 -2 -3 -6 0 -1 2 -13 -10 6 1 14 0 5 -7 13 -3;
-1 13 10 17 -4 10 -2 3 0 0 -12 10 1 8 6 -1 -3 -3 10 7;
-10 -7 15 5 -5 7 -3 4 -4 13 -5 -18 -5 -10 5 4 17 -16 -3 2;
4 6 -8 0 3 18 9 19 -12 4 -13 -4 -16 5 18 0 15 -5 1 10;
3 -3 8 14 5 -16 6 1 0 -2 7 13 10 -7 -2 -6 3 2 7 14;
16 -2 0 -1 14 -3 -6 13 -1 -13 15 -11 4 -4 -5 -13 -13 -10 -1 3;
-1 -2 6 -16 3 3 4 -5 -2 -2 12 -7 0 3 5 7 -8 5 17 14;
-6 -6 2 4 -11 5 -8 11 -11 2 -7 1 9 -8 -11 1 2 -3 -2 6;
-4 1 -1 3 -13 -10 -8 -1 10 -11 -12 10 5 18 -4 8 12 -10 8 -7;
-5 14 2 5 -5 -1 -11 -4 -8 1 -7 -17 18 -4 -12 -19 5 5 6 -2;
15 -13 14 12 13 3 -7 -9 -1 16 -6 11 -13 -12 -7 -8 3 -5 -12 7;
14 -2 13 3 8 6 -11 4 -8 -14 13 17 3 -13 -5 11 -11 -5 -4 5;
-11 -10 2 -7 17 -5 1 -11 0 5 11 10 -4 10 -17 6 10 -18 10 13;
-1 -1 6 -2 -8 -2 -11 16 -14 -6 -1 0 -12 10 -8 3 -1 -4 0 0;
-13 2 11 -2 -14 6 2 2 -6 14 16 5 4 -11 1 20 11 13 0 -2;
-13 0 -11 7 11 2 1 -2 3 20 -8 6 0 8 -19 8 7 4 -1 -6;
-3 -3 0 3 6 -18 5 1 -2 6 3 -5 18 -10 -1 2 -4 7 10 -16;
-13 5 -2 -8 -11 -4 2 -15 -1 11 3 10 15 12 5 7 0 17 -3 3;
0 -12 -4 6 13 0 2 9 6 11 14 2 -8 -1 2 -11 -2 15 10 8];
To produce symmetrizable random matrices A of arbitrary size (here: n=20), you can use
n = 20;
C = randi([-10 10],[n n]);
B = (C+C.');
Per = eye(n);
Per = Per(randperm(n),:);
A = Per*B;

R2021b

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