Question: Positive-Definite Correlation Matrix

I am trying to build a portfolio optimization model from simulated data. I start by defining the very large

correlation matrix and then I apply Cholesky decomp to get the cross correlated simulated data. However

I systematically get the warning message:  Warning, Matrix is not positive-definite. It is extremely difficult

(if possible at all) to specify a large correlation matrix that is positive-definite.

 

I read that the eigenvalues need to be positive. So my question is how can I force the large correlation

matrix to be positive-definite. There must exist some way that you can build up the correlation matrix

step-by-step to ensure this.

 

For example consider this example:

 

First of all we know that the correlation matrix is symmetrical so we only need to worry about the

righthand side of the diagonal.  Secondly we know that all the diagonal correlations are 1 .

Thirdly there appears to be "super symmetry" in a sense that we only need to specify 3 correlations out of 6.

Again there must exist a way to take advantage of this to build a positive-definite matrix step-by-step....?

 

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