Clearer explanation to get to Eq. 10.21

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Describe the mistake
This is not a mistake per se; I do have a EE degree but am not a Math major, so self-taught in Math for ML, and

I spent a few days struggling to prove to myself the below equation step in the paragraph just before Equation 10.21.

(S − B_{m−1} S) b_i = Sb_i

found in the sentence:
In the case i ⩾ m, by using (10.20) in (10.19b), we obtain ˆS bi = (S −
Bm−1S)bi = Sbi = λibi, ...

Location
Please provide the

1. version pdf, Draft (2024-01-15)
2. Chapter 10
3. page 324
4. line number/equation number line number 5, paragraph above equation 10.21

Proposed solution
Show the intermediate steps:
(S − B_{m−1} S) b_i
= S b_i - B_{m-1} S b_i
= S b_i - B_{m-1} λ_i b_i
= S b_i - λ_i B_{m-1} b_i
= S b_i - λ_i (sum_{j=1}^{m-1} b_j b_j^T) b_i
= S b_i - λ_i (sum_{j=1}^{m-1} b_j 0) because b_j^T and b_i are orthogonal since i ⩾ m
= S b_i
= λ_i b_i

Additional context
N/A
P.S. I enjoy and would recommend the book so far, and appreciate the care and extra explanations that I could see have gone into the book so far.