can you please tell me why eigenvalues are used in PCA. Specifically why and how does it explain the variance of the components
I believe that PCA is one of the most well-suited applications of eigenvector/eigenvalues.
- Raw data can possibly be coupled(made complex) with principal component.
- Some coupling can be thought of linear transformations.
- Eigenvalues and eigenvectors can be used to decouple linear transformation.
- Calculate the accuracy of mapping to the new set of data.
- Let's package it nicely with a name called PCA.
I found a good youtube video that explain PCA well.
To explain coupling, think of two variables, area,$A$ and perimeter,$P$ of a rectangle. We know that $A = width * height$ and $P = 2(width+height)$. Thus we can say that $A$ and $p$ is coupled with $width$ and $height$ of the rectangle, changing $A$ will also change $p$ because of the change of $width$ and $height$. Though the example I given is not linear, you can get the idea.