# Determine missing eigenvalues given only correlations between variables and components

Given that I just have a correlation matrix ($$X$$ Variables vs. $$Y$$ Principal Components), and that I am trying to find 2 missing eigenvalues (e.g., missing $$\lambda_1$$ and $$\lambda_5$$) from the total $$X$$ eigenvalues. How should I proceed?

• Not sure I understand the set up. Do you have an n by n correlation matrix and know some of the eigenvalues of the covariance matrix? Take for instance the correlation matrix being the 2 by 2 identity matrix with one of the eigenvalues being known, if the smaller eigenvalue is known, the larger eigenvalue could be any number $\ge$ that. If the larger eigenvalue is known, the other can be any number $\le$ that and $\ge 0$. Jun 15 '19 at 23:36

If you have the remaining eigenvales, then you can try to interpolate. This will not work for missing $$\lambda_1$$, of course, but may work well for other eigenvalues.
You can try using Marchenko-Pastur distribution like in this answer. This distribution concerns eigenvalues of random matrices, particularly, large ones. I would think it can help filling the blanks by providing you with a shape of eignevalues spectrum.