Why is the sum of squared coefficients of x's across all the Principal Components 1

pca

#1

hello,

this is the loadings of pca .In here the length of each Eigen Vector is 1 as that is one of the restrictions applied while performing PCA,but the squared sum of each x’s coefficients across all the components is also 1:

> pca1$loadings[2,1]^2+pca1$loadings[2,2]^2+pca1$loadings[2,3]^2+pca1$loadings[2,4]^2+pca1$loadings[2,5]^2+pca1$loadings[2,6]^2+pca1$loadings[2,7]^2+pca1$loadings[2,8]^2+pca1$loadings[2,9]^2+pca1$loadings[2,10]^2+pca1$loadings[2,11]^2+pca1$loadings[2,12]^2+pca1$loadings[2,13]^2
[1] 1
> pca1$loadings[1,1]^2+pca1$loadings[1,2]^2+pca1$loadings[1,3]^2+pca1$loadings[1,4]^2+pca1$loadings[1,5]^2+pca1$loadings[1,6]^2+pca1$loadings[1,7]^2+pca1$loadings[1,8]^2+pca1$loadings[1,9]^2+pca1$loadings[1,10]^2+pca1$loadings[1,11]^2+pca1$loadings[1,12]^2+pca1$loadings[1,13]^2
[1] 1

Is this because each PC captures a part of the variance of x’s through the eigen vectors and since the variables are standardized the variance of each x is 1??
Also how do I extract the Eigen values from the PCA’s?


#2

@data_hacks-yes your reason is correct the PC captures the a part of the standardized variance.

You can access the Eigen through the values of rotation in the result.

Hope this helps!
Regards,
hinduja1234