make an r by m permutation matrix

I have the following code :

A = eye(r);
L = [];
for i = 1:10
    A = A(randperm(r),:);
    L = [L A]; % 10 x 100 matrix with only one nonzero entry per column
end

The variable L is a 10x 100 size matrix. however i want to make a more intelligent code such that,

the dimensions will depend on the dimension of an input data matrix. for example when i input X, i.e.

% r is my rank. m and n are the dimensions.
X = rand(m,r)* rand(r,n);

% I want L to have dimension of [m by r] or [r by n] .
%  i.e. it should pick on of the sides of X and the other side r 

Answer 1

[Parag]

Hi, the below code generates the dimensions m, n, and r randomly, ensuring that r is always less than or equal to the minimum of m and n. It then constructs a matrix X as the product of two random matrices of sizes (m×r) and (r×n). Based on the dimensions of X, the function “generate_L “ creates a matrix L of size either (m×r) or (r×n), using randomly permuted identity matrices. This ensures L has only one nonzero entry per column while adapting to the input matrix X.

Finally, the code outputs the randomly chosen values of m, n, and r, along with the size of L, making the entire process fully automated and flexible.

function L = generate_L(X, r)
    [m, n] = size(X);
    
    % Determine the size of L based on X
    if m >= n
        L = generate_permutation_matrix(m, r); % Size [m x r]
    else
        L = generate_permutation_matrix(r, n); % Size [r x n]
    end
end

function L = generate_permutation_matrix(rows, cols)
    L = [];
    I = eye(rows);
    for i = 1:cols
        permuted_I = I(randperm(rows), :); % Random permutation of identity
        L = [L, permuted_I];  % Construct the final matrix
    end
end

m = randi([5, 15]); 
n = randi([5, 15]); 
r = randi([min(m, n)-2, min(m, n)]); % Random r ensuring 1 <= r <= min(m, n)

X = rand(m, r) * rand(r, n);

L = generate_L(X, r);

disp(['m = ', num2str(m), ', n = ', num2str(n), ', r = ', num2str(r)]);
disp('Size of L:');
disp(L)
disp(size(L)); % Will output either [m x r] or [r x n]