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mml-book / mml-book.github.io

Companion webpage to the book "Mathematics For Machine Learning"
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Ask to add more explanation in 4.5 Singular Value Decomposition #641

Open nijkah opened 3 years ago

nijkah commented 3 years ago

I am using the pdf of 2021/03/28.

In equation 4.77 and 4.78 I think it is not clear where the lambda come from.

In equation 4.77, (A^\top A) v_j is converted to (\lambda_j v_j). It may be derived from equation 4.54 ( A p_n = \ambda_n p_n ).

In equation 4.78, actually it is obvious that \sigma_i is derived from 4.75. But I am confusing how the || A v_i || is converted to \sqrt{\lambda}.

Readers of this book will appreciate it if you provided more clear explanation. If it is trivial, I'll try to blame my head. :(

mpd37 commented 3 years ago

By definition: || A v_i ||^2 = (Av_i)^T (Av_i). This is written out in (4.77), if you choose v_i = v_j. Take the square root because we don't divide by the squared norm.

Does this make sense now?

I'll mark this as a feature request: We can't make this change right now, but it'll be good to have that in the future.