### Factor analysis: Existence of solution to factor models

*Adrià Prior Rovira*

#### Abstract

One of the main results in factor analysis states that if the factor model holds for a random vector X, then the covariance matrix of the vector admits a decomposition in terms of the matrices that characterize the model, and that the converse is also true under quite general conditions. The direct implication can be found proved in many references but the proof of the converse seems diffcult to find in the available texts. The reason of this article is to share an original proof of the converse, first for the case of the orthogonal factor model, widely used in exploratory factor analysis, and secondly for a factor model that generalizes the orthogonal one, and which is meant to be used in confirmatory factor analysis.

Keywords: factor analysis, exploratory factor analysis, confirmatory factor analysis, factor model, fundamental theorem of factor analysis.

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ISSN: 2385-4227 (electronic edition)