How to generate a random positive semi-definite matrix of certain size with real numbers in a certain range?

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Khanh
Khanh 2011-10-21
I'm looking for a way to generate a *random positive semi-definite matrix* of size n with real number in the *range* from 0 to 4 for example.
I didn't find any way to directly generate such a matrix. However, I found that *Lehmer* matrix is a positive definite matrix that when you raise each element to a nonnegative power, you get a positive semi-definite matrix.
So, I did something like this
A=16*gallery('lehmer',100) %matrix of size 100*100 in range 0-16
B=A.^(1/2) %scale down to range 0-4
So my questions are:
1. I wonder if that maintains the randomness of the matrix?
2. Is there any direct way to generate random positive semidefinite matrix?
Thanks,

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Walter Roberson
Walter Roberson 2011-10-21
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Khanh
Khanh 2011-10-26
The problem with the solutions in the thread that you referred to is that you have C=A*A'. (C is the result here)
So if you want C[i][j] to be in a range say 0 to 5. Then you need A=sqrt(5)*rand(n).
Then the result C will not truly random in the range 0 to 5, I think.
Walter Roberson
Walter Roberson 2011-10-26
Your question as phrased was about constructing random PD matrices whose entries were all within a certain range.
That is a different matter than constructing matrices whose entries have a uniform distribution within a given range and the matrix as a whole is PD. Such matrices appear to be relatively rare to start with; to require uniform random distribution on the entries appears to make them difficult to generate.

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