linear algebra - Algorithm for generating positive semidefinite matrices

I'm looking for an efficient algorithm to generate large positive semidefinite matrices.

One possible way I know of is:

• generate a random square matrix


• multiply it with its transpose.


• calculate all eigenvalues of the result matrix and check if all of them are non-negative.

However, this approach is infeasible given a large matrix, say $1000 \times 1000$ or more.

Could anyone please suggest an efficient way to generate a positive semidefinite matrix?
Is there any MATLAB function for this job?

Thanks,

Answer

Any matrix multiplied by it's transpose is going to be PSD; you don't have to check it. On my computer raw Octave, without SSE, takes 2 seconds to multiply a 1000x1000 matrix with itself. So not all that infeasible.

If you don't like that, you can always just generate a random diagonal matrix. That's sort of the trivial way, though :) What do you need the matrix for? If it's as test input to another algorithm, I'd just spend some time generating random PSD matrices using the above matrix-matrix multiplication and save the results off to disk.