如何在二进制矩阵中找到最可靠的基？

Question

我想编写一个允许执行以下指令的 MATLAB 程序，但是我遇到了一些困难，所以如果有人帮助我，我将非常感激。

设一个二进制矩阵A。从A的第一列开始，找到前K个线性独立的列，关联可靠性值最大。然后，将这 K 个线性独立的列用作新矩阵 B 的前 K 列，保持它们的可靠性顺序。 B 的其余 (N - K) 列也按可靠性降序排列。

例子：

A = [1 0 0 1 0 0 1 1;
     0 1 0 1 1 0 0 1;
     0 0 1 1 1 0 1 0;
     0 0 0 0 1 1 1 1]

A的前三列线性独立，第五列与前三列线性独立。

我们发现：

B = [1 0 0 0 1 0 1 1;
     0 1 0 1 1 0 0 1;
     0 0 1 1 1 0 1 0;
     0 0 0 1 0 1 1 1]

Answer 1

您可以使用rank 来确定列是否线性独立，如下所示：

K = size(A, 1); % the number of linearly independent columns to find
B = zeros(size(A)); % preallocate output for efficiency
colB = 1; % column index for the next linearly independent column
colBafter = K + 1; % column index for the next linearly dependent column

for i=1:size(A, 2) % loop over all columns in A
  B(:, colB) = A(:, i); % add the next column of A as an independent column to B
  if rank(B(:, 1:colB)) == colB % check if the newly added column is indeed linearly independent
    if colB == K % check if all independent columns are found
      break;
    else
      colB = colB + 1; % increase the independent index as the added column was indeed linearly independent
    end
  else
    B(:, colBafter) = A(:, i); % add the column as a dependent column to B
    colBafter = colBafter + 1; % increase the dependent index as the added column was linearly dependent
  end
end

B(:, colBafter:end) = A(:, i+1:end); % copy the rest of A to B