What are eigenvalues and eigenvectors in factor analysis?

I understand the idea behind factor analysis, but everything I read on the topic seems to very vaguely cover the topic of eigenvalues and eigenvectors

Whats the correct way to understand eigenvalues and eigenvectors in factor analysis? What are they? and how are they related to factors, communalities, variance capture, and factor loadings?

I'm not so much interested in how we decompose a matrix into eigenvalues and eigenvectors, but rather how we interpret them in the context of factor analysis

This becomes especially important when employing the Kaiser rule (eigenvalues > 1) and looking at scree plots (where the Y axis is eigenvalue)

Ive seen similar questions about this (Eigenvalue vs Variance and What is the rationale behind the "eigenvalue > 1" criterion in factor analysis or PCA?), but the answers dont really explain the link between the 2 concepts and the ideas/language of factor analysis

EDIT:

I saw on this site that "the eigenvectors of R (multiplied by their eigenvalues) are known as the factor loadings and are literally the correlations of the each variable in X with an underlying factor or principal component" ... Is there an intuitive way to understand why this is the case?

• We speak of eigenvalues and eigenvectors (of the correlation/covariance matrix) in PCA . In FA, we may speak of them in principal axis method of extraction (PAF). See stats.stackexchange.com/q/205459/3277 for clarification. See stats.stackexchange.com/q/102882/3277 comparing PCA and PAF FA of iris data. – ttnphns Oct 10 '17 at 6:34
• The first link helps somewhat, but it doesnt quite provide the full picture I'm looking for. What is the difference between a factor and a principal component (I know how theyre calculated, but whats the conceptual difference)? When you talk about eigenvalues representing variance of a factor/PC, is that similar to what a communality is telling you (variance in the item captured by the set of factors)? And how do eigenvectors fit into this picture? – Simon Oct 10 '17 at 22:53