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.

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.

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