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    <p><a href="http://en.wikipedia.org/wiki/Kernel_%28matrix%29#Numerical_computation_of_null_space" rel="nofollow">This</a> Wikipedia article describes three methods for the numerical computation of the null space: reduction (Gaussian elimination), <a href="http://www.mathworks.com/help/techdoc/ref/svd-singular-value-decomposition.html" rel="nofollow">SVD</a>, and <a href="http://www.mathworks.com/help/techdoc/ref/qr.html" rel="nofollow">QR decomposition</a>. In brief, (1) reduction is "not suitable for a practical computation of the null space because of numerical accuracy problems in the presence of rounding errors", (2) SVD is the "state-of-the art approach", but it "generally costs about the same as several matrix-matrix multiplications with matrices of the same size", and (3) the numerical stability and the cost of QR decomposition are "between those of the SVD and the reduction approaches". </p> <p>So if SVD is too slow, you could give a chance to <a href="http://www.mathworks.com/help/techdoc/ref/qr.html" rel="nofollow">QR decomposition</a>. The algorithm with your notations is as follows: "<code>A</code> is a <code>4xN</code> matrix with <code>4&lt;N</code>. Using the QR factorization of <code>A'</code>, we can find a matrix such that <code>A'*P = Q*R = [Q1 Q2]*R</code>, where where <code>P</code> is a permutation matrix, <code>Q</code> is <code>NxN</code> and <code>R</code> is <code>Nx4</code>. Matrix <code>Q1</code> is Nx4 and consists of the first 4 columns of <code>Q</code>. Matrix <code>Q2</code> is <code>Nx(N-4)</code> and is made up of the last <code>N-4</code> columns of <code>Q</code>. Since <code>A*Q2 = 0</code>, the columns of <code>Q2</code> span the null space of <code>A</code>." </p> <p>Matlab implementation: <code>[Q, R, P] = qr(A', 'matrix');</code> The columns of matrix <code>Q2 = Q(:, 5:end);</code> give the null space of <code>A</code>.</p>
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    1. POObtain null space or single dimensional space which is its best approximation efficiently
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    1. COMakes sense.. Although, isn't there a way to get create another matrix using A in which the singular values would be reversed? I could just call svds after that and get only the first or the second columns... That could be faster than this. Anyways, I just sampled coz N was too large anyways. Thanks for your reply.
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