# [R] Positive Definite Matrix

Mike Miller mbmiller at umn.edu
Sun Jan 30 18:28:48 CET 2011

```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

```