Message-ID: <alpine.DEB.2.00.1101301123500.13255@taxa.psych.umn.edu>
Date: 2011-01-30T17:28:48Z
From: Mike Miller
Subject: Positive Definite Matrix
In-Reply-To: <C2226081-3F61-444B-ADFA-6CCB6FE3B9A8@comcast.net>
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