# Outer Product of leading eigenvector with itself

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.

## 1 Answer

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 $$v_{k+1}=\frac{Mv_k}{\|Mv_k\|}$$

• Thanks for the answer. That makes it more clear. Can you please expand on how is d||y|| = (y^T dy) / ||y|| ? – user2125722 May 27 at 22:01
• This of course only holds for the Euclidean norm, $\|y\|=\sqrt{y^⊤ y}$, where the outer derivative is $\frac1{2\sqrt{..}}$ and the inner derivative is $2y^⊤dy$ – Dr. Lutz Lehmann May 27 at 22:12