rank approximation in matlab

To rank an approximation in MATLAB, we can use the singular value decomposition (SVD) method along with the least squares method.

First, we can generate a matrix with some random values. For example:

main.m
A = rand(100, 50);
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Next, we can calculate the SVD of this matrix:

main.m
[U, S, V] = svd(A);
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The singular values are stored in the diagonal matrix S. We can plot these values to determine the rank of the matrix:

main.m
plot(diag(S), 'o');
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The rank of the matrix can be determined by finding the number of significant singular values. We can determine the threshold by inspecting the plot and visually determining where the significant values end:

main.m
threshold = 1e-10; % adjust this value as needed
rank_A = sum(diag(S) > threshold);
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Finally, we can calculate the best rank-k approximation of the matrix using the least squares method:

main.m
k = 10; % desired rank
A_approx = U(:,1:k) * S(1:k,1:k) * V(:,1:k)';
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This approximation will have rank k, and will be the best approximation in terms of the least squares method.