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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.

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

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  • $\begingroup$ Thanks for the answer. That makes it more clear. Can you please expand on how is d||y|| = (y^T dy) / ||y|| ? $\endgroup$ – user2125722 May 27 at 22:01
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    $\begingroup$ 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 $2y^⊤dy$ $\endgroup$ – Dr. Lutz Lehmann May 27 at 22:12

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