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In PCA, we essentially create a new factor with the original variables in a linear combination manner such that: $Y_1 = a_{11} \times x_1 + a_{12} \times x_2 + a_{1p} \times x_p$.

The sum of squared loadings of all variables (features) for each PC sum to 1. Therefore, is it safe to say that factor $i$ contributes the squared loading amount to the PC $j$?

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According to this presentation:

Factor loadings (factor or component coefficients) : The factor loadings, also called component loadings in PCA, are the correlation coefficients between the variables (rows) and factors (columns). Analogous to Pearson's r, the squared factor loading is the percent of variance in that variable explained by the factor. To get the percent of variance in all the variables accounted for by each factor, add the sum of the squared factor loadings for that factor (column) and divide by the number of variables. (Note the number of variables equals the sum of their variances as the variance of a standardized variable is 1.) This is the same as dividing the factor's eigenvalue by the number of variables.

Eigenvalue: Column sum of squared loadings for a factor, i.e., the latent root. It conceptually represents that amount of variance accounted for by a factor. The ratio of eigenvalues is the ratio of explanatory importance of the factors with respect to the variables. If a factor has a low eigenvalue, then it is contributing little to the explanation of variances in the variables and may be ignored as redundant with more important factors. Eigenvalues measure the amount of variation in the total sample accounted for by each factor. A factor's eigenvalue may be computed as the sum of its squaredfactor loadings for all the variables.

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