Why is the first eigenvalue on the principal component always superior to the second eigenvalue?

Question

I'm learning PCA and I wanted to know why the first eigenvalue on the principal component is always superior at second eigenvalue. Thanks for helping me

Ps: I tried to unterstand before asking this question but to be honest, I had difficult to unterstand what I was reading on book. I would like just an easy answer who can allow me to unterstand it better

Answer 1

This is the way it is conventionally presented, and for good measure, since the higher eigenvalues explain a higher percentage of the variance in the dataset. This is sometimes reflected in PCA analysis in what is called a scree plot, depicting the percentage of the variance accounted for by each of the eigenvalues ordered from higher to lower, and with the intent of summarizing the data selecting only the first (highest) eigenvalues, which explain the majority of the variance with reduced dimensionality.

Here is an intuitive graphical representation of what we try to achieve by selecting the eigenvectors of the largest eigenvalues of the covariance matrix: