First, let's do an initial approximation... a 1 million by 1 million matrix (let's call it X) has 1,000,000,000,000 elements.  Depending on the type of data that you are storing, each element could take a certain number of bytes. If you save the matrix as double (for example) and then convert it to sparse the size could be ~ 2,400,080,000 bytes...

Check out this post at Stackoverflow (handling a very big sparse matrix in Matlab), maybe it could help you with your particular applications.

http://stackoverflow.com/questions/24789600/handling-a-very-big-and-sparse-matrix-in-matlab.

Also remember that if you are doing SVD then X = U*S*V', so you will also need memory for the matrix operations and to store U, S and V. (I haven't used svds but you get the idea....)

Hope this helps,

Carlos

Dear Mr. Chan,

Good day,

attached are some links and articles that are related to the subject, please read the pdfs first and see the rest after that and try and solve your problem, if you have some difficulties, you can come back:

http://uk.mathworks.com/matlabcentral/fileexchange/11120-null-space-of-a-sparse-matrix

https://www.mathworks.com/matlabcentral/newsreader/view_thread/249467

http://stackoverflow.com/questions/18486101/finding-null-space-of-a-large-sparse-matrix-in-matlab

https://www.quora.com/How-could-you-manipulate-sparse-matrix-in-MATLAB

http://www.scriptscoop.net/t/0741a363f669/finding-null-space-of-a-large-sparse-matrix-in-matlab.html

http://troublevn.com/153095/how-to-compute-null-of-large-sparse-matrix-in-matlab

https://en.wikipedia.org/wiki/Lanczos_algorithm

@Lyle Actually I just want my result to be the actual null space, which is around a 1000000x1000000 size matrix.

@Hazim I actually found a null space computation, except that it can run for days. Is there a possible way also for shortening run-time?

@Carlos Memory mapping is fine, but wouldn't I lose data when I use the decomposition approach?