Creating random binary 3D matrix with specified rank

I need to create k number of n by m binary matrixes (lets call each of them Hᵢ where i from 1 to k) where for each of these matrixes their rank should be equal to m. Therefore, rank(Hᵢ) = m.

I think I can create them by using (not sure):

function [Y,rk] = generateMatrix(w,y,rank)
P = orth(randi([0 1], w, rank));
Q = orth(randi([0 1], y, rank))';
Y = P*Q;
rk = rank(Y);
end

Now, I want to achieve rank(H꜀) = min(n, k*m) where H꜀ = [H₁ H₂ ... Hₖ].

How can I achieve this?

Thanks

Answer 1

[Gayatri Rathod]

Hi Efe Berkay,

• To achieve a rank of Hc equal to min (n, k*m), where Hc is the horizontal concatenation of the k binary matrices H₁, H₂, ..., Hₖ, you can modify the generateMatrix function to generate matrices with a desired rank as follows:

 function [Y,rk] = generateMatrix(w,y,rank)
    P = orth(randi([0 1], w, rank)); 
    Q = orth(randi([0 1], y, rank))'; 
    Y = P*Q; 
    Y = Y (:, 1:min(size(Y)));  % truncate to desired rank 
    rk = rank(Y); 
 end 

• This modified function generates a binary matrix Y of size w by y with rank equal to the specified rank. Then, it truncates the matrix Y to the first min (w, y, rank) columns, ensuring that the resulting matrix has the desired rank.
• To generate k matrices H₁, H₂, ..., Hₖ with rank equal to m, you can call the modified generateMatrix function in a loop:

 n = 10; % size of each matrix 
 m = 3; % desired rank of each matrix 
 k = 4; % number of matrices to generate 
 H = zeros(n, m*k); % initialize H 

 for i = 1:k 
     % generate a matrix Hi with rank m 
     [Hi, ~] = generateMatrix(n, m*k, m); 

     % add Hi to H 
     H(:, (i-1)*m+1:i*m) = Hi; 
 end 
 
 % check rank of H 
 rk = rank(H); 
 rk_desired = min(n, k*m); 
 fprintf('Rank of H: %d (desired rank: %d)\n', rk, rk_desired); 

• This code generates k binary matrices H₁, H₂, ..., Hₖ with rank equal to m, and concatenates them horizontally to form the matrix H. The rank of H is then checked to ensure that it is equal to the desired rank of min(n, k*m).

You can read more about the rank function from the following documentation: rank Function.

Hope it helps!

Regards,

Gayatri Rathod