https://doi.org/10.1351/goldbook.10093
Matrix decomposition of a data matrix (\(\boldsymbol{X}\)) into the product of a scores matrix (\(\boldsymbol{T}\)) and the transpose of the loadings matrix (\(\boldsymbol{P}^{\rm{T}}\)).
Notes:
- Hence \(\boldsymbol{X} = \boldsymbol{TP}^{\rm{T}} + \boldsymbol{E}\), where \(\boldsymbol{E}\) is a residual matrix.
- Factor analysis methods include common factor analysis (also called ‘factor analysis’), principal component analysis, and multivariate curve resolution.
- The number of factors selected in factor analysis is smaller than the rank of the data matrix.
- Factor analysis is equivalent to a rotation in data space where the factors form the new axes. This is not necessarily rotation that maintains orthogonality except in the case of PCA.
- The residual matrix contains data that are not described by the factor analysis model, and is usually assumed to contain noise.