Outer Product of leading eigenvector with itself

Question

I came across the following equation while reading a paper. Can anyone please explain me the how is the first step obtained ? Here, $v_{k+1}$ is the leading eigenvector computed for the matrix $M$ using the power iteration method. Any hints would be really helpful. Thanks.

Answer 1

Essentially you are asking about the derivative of $\frac{y}{\|y\|}$. By chain rule, $$ d\frac{y}{\|y\|}=\frac{dy}{\|y\|}-\frac{y(y^\top dy)}{\|y\|^3}=\frac{I-\frac{yy^T}{\|y\|^2}}{\|y\|}dy $$ Now set $y=Mv_k$ and use that $v_{k+1}=\frac{Mv_k}{\|Mv_k\|}$