Positive Definite Matrix
On Sun, 30 Jan 2011, David Winsemius wrote:
On Jan 30, 2011, at 6:02 AM, Alex Smith wrote:
Thank you for all your input but I'm afraid I dont know what the final conclusion is. I will have to check the the eigenvalues if any are negative. Why would setting them to zero make a difference? Sorry to drag this on.
The discussion is proceeding on the assumption that the "true" matrix is PD and that only because of numerical imprecision has a negative eigenvalue been reported. You would only decide to set the negative eigenvalues to zero if you had prior knowledge that the matrix _should_ be PD and that you needed to so something further with the matrix on that basis. Usually the matrices in question are the result of many calculations that may have introduced sufficient numerical round-off error to distort the result.
In one common scenario you have computed variances and covariances individually, then constructed a var-covar matrix from them. When the true var-covar matrix was nearly singular, a matrix estimated in this way can be negative definite because of different patterns of missing values for different pairs of variables. All true var-covar matrices are non-negative definite: They may be singular (having at least one zero eigenvalue), but they cannot have a negative eigenvalue. Mike