Emil O W Kirkegaard

A non-positive definite correlation matrix means that the interrelationship of variables in the matrix is not mathematically possible. (It's not unlike a GRIM test showing that a mean is not possible, given n.) It means something's wrong with this data summary. 1/2

It might be as innocent as using pairwise deletion (where some correlations in the table have a slightly different n due to missing data) or a typo. Regardless of the reason, it shows that the reporting is deficient in some way. 2/2

Can this be explained by rounding? That is, could the matrix become positive definite if .12 were replaced by .12256, etc? (I've never encountered someone using this technique before; it sounds potentially promising)

Not with SPSS. It program takes rounding into consideration. In my pre-print where I discovered this anomaly, I factor analyzed 99 correlation matrices. 97 were positive definite. If rounding were regularly a problem, then I would have had more than 2 problem datasets.

Is that necessarily true? I don't know enough about the properties of matrices, but it seems to me that there must be a border, and perhaps only a few are close to it. (Mostly just being devil's advocate here.)

I'm not entirely sure. It might be specific to SPSS. The fact that a 50-variable matrix (from Guthrie, 1963) that was rounded to 2 decimal places worked is a strong indication that the program can handle rounding. (1/2)

50 variables in a matrix = 1,225 unique matrix cells. The more values there are in a matrix, the harder it is for a matrix to be positive definite if there is a problem with the data. (2/2)

I guess my question is how many values would need to be changed (and/or by how much) for it to become positive definite. I'm not immediately convinced by the argument that if one 50-item matrix works with rounding, that proves the method is generally robust to rounding.