Model of a matrix

How can I model the matrix of full permutation [(1,1,0,...,0) (-1,-1,0,...,0)] for n variables.

Answer 1

[BhaTTa]

To model a matrix of full permutation with (n) variables where each row is a permutation of ((1, 1, 0, \ldots, 0)) and ((-1, -1, 0, \ldots, 0)), you can follow a systematic approach. Since you're interested in permutations with exactly two non-zero elements (either both (1)s or both (-1)s) and the rest zeros, the number of such permutations for each case ((1)s or (-1)s) is (\binom{n}{2}), which is the number of ways to choose 2 positions out of (n) for the non-zero values.

Here's how you can achieve this in MATLAB:

Step 1: Generate the Basic Permutation Vectors

First, generate a basic vector for the (n) variables, which will be ([1, 1, 0, \ldots, 0]) and ([-1, -1, 0, \ldots, 0]), then find all permutations where exactly two positions are non-zero.

Step 2: Create a Function to Generate Unique Permutations

You need to generate all unique combinations of positions for the two non-zero values in the vector of length (n). Each combination will correspond to a row in the final matrix.

MATLAB Code Example

Here's a MATLAB function that generates the desired matrix for (n) variables:

function fullPermMatrix = generateFullPermutationMatrix(n)
    % Initialize the matrix to hold the permutations
    % There are n choose 2 combinations for each of 1s and -1s
    numPerms = nchoosek(n, 2);
    fullPermMatrix = zeros(2*numPerms, n);
    
    % Generate combinations for positions of non-zeros
    comb = nchoosek(1:n, 2);
    
    % Fill the matrix with permutations for 1s
    for i = 1:size(comb, 1)
        % For 1s
        fullPermMatrix(i, comb(i, :)) = 1;
        % For -1s (starting after the 1s permutations)
        fullPermMatrix(i + numPerms, comb(i, :)) = -1;
    end
end

How to Use This Function

Call this function with your desired (n) to get the matrix:

n = 4; % Example for n = 4
fullPermMatrix = generateFullPermutationMatrix(n);
disp(fullPermMatrix);

Output

For (n = 4), the output will be a matrix where each row is a unique permutation of ([1, 1, 0, 0]) and ([-1, -1, 0, 0]) as per the specified criteria. The size of this matrix will be (2 \times \binom{n}{2}) by (n), reflecting all permutations of placing two (1)s and two (-1)s among zeros.

This approach ensures that you get a compact representation of all the required permutations for your problem.