Simplest way of creating a positive (semi-) definite matrix?

Question

I want to construct an arbitrary $n\times n$ positive semi-definite matrix.
Maybe I can randomly create an $n\times m$ matrix and use its covariance matrix.
But I want to know if there's any other (efficient) way of doing this?

Thanks.

The problem is not trivial. See these questions on MSE and SO: a, b, c. See also this article on random orthogonal matrices — Jean-Claude Arbaut, Mar 31 '16 at 09:39

score 3 · Accepted Answer · answered Mar 31 '16 at 07:26

3

You can try if this way works out for you:

Create a diagonal Matrix $M$ with diagonal entries $(\geq) > 0$. Then take an invertible matrix $P$ and calculate $E = PMP^{-1}$, hence E and M are similar, hence E is also positive (semi-) definit. Using this way you have to worry about finding arbitrary invertible matrices. One way for that would be to do random elementary row/column operations to the identity matrix(for that you can create elementary matrices by random, which is an easy task).

answered Mar 31 '16 at 07:26

CandyOwl

261

you're right thank you – CandyOwl Mar 31 '16 at 08:26
what about using orthognal matrices $P$? One could generate them using gramm schmidt agorithm – CandyOwl Mar 31 '16 at 09:24

Simplest way of creating a positive (semi-) definite matrix?

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