How to parameterize the positive semi-definite correlation matrix?

From: T. Wakamatsu (WakamatsuT_at_pac.dfo-mpo.gc.ca)
Date: 12/10/04 

Date: Fri, 10 Dec 2004 19:12:52 +0000 (UTC)

Hi there

I am trying to compute random vectors obeying a certain correlation
matrix parameterized by the following function.

C(x_i,y_i;x_j,y_j)=exp(-(x_i-x_j)**2/L-(y_i-y_j)**2/L)

Where, L is the decorrelation length scale and (x_i,y_i) is
a point on a 2D plane. So, I am trying to construct 2D random field.
The eigenvalue-eigenvector decomposition of the correlation matrix
works in perfect to construct square root of the matrix in a sample
problem with smaller size, say 10*10 matrix. But when I apply
the same procedure for my real problem of 10000*10000 matrix,
it starts to reply negative eigen values.
Questions are:
1. Does the above function not guarantee the positive
semi-difinitiness of the resultant correlation matrix?
2. If not, is there better simple function to parameterize
a correlation matrix?

Thanks

Tsuyoshi