PCA not showing Eigenvalues for all variables in a correlation matrix

Question

Hello,

I have a correlation matrix (not the raw data of a dataset) which is a 7x7 matrix of seven variables.  I put all 7 of the variables into the Y column of the Principal Components menu and JMP will generate a list of eigenvalues.  However, it is only generating SIX eigenvalues instead of all seven.  For some reason one of the 7 variables that was put into the Y is not getting included in the analysis. Similarly, the Scree Plot is only showing 6 of the 7 variables.

Any idea what I'm doing wrong here?  My understanding is that all 7 of the variables should be getting eigenvalues and a spot on the Scree plot.

Thanks!

ih · Accepted Answer

A correlation matrix has one less degree of freedom than the number of columns in the dataset, so in a multivariate model you should expect at least one fewer component than you have columns.

Consider the example below, you can fill in the values for any column or row by looking for the same value somewhere else in the matrix (the top right and bottom left triangles are the same).  Thus, as @Dan_Obermiller pointed out, all of the variation in the matrix can be explained by at least one fewer component than you have columns.  If some variables can be fully explained by others, then you could see even fewer eigenvalues/factors.