The problem is to show that, given norm(y)_2=lambda^T y,norm(λ)_2 <= 1 and y != 0, we have lambda=(y)/(norm(y)_2).

odcinaknr 2022-10-05 Answered
The problem is to show that, given y 2 = λ T y , λ 2 1 and y 0, we have λ = y y 2
My approach is, y 2 = | λ T y | y 2 λ 2 λ 2 1 which combined with λ 2 1 gives that λ 2 = 1. So λ and y are not oppositely aligned, since y 2 0
Also, y 2 = λ T y ( y y 2 λ ) T y = 0. But since we showed that λ and y are not oppositely aligned, this should mean that the only possibility is y y 2 λ = 0 which gives the result.
I feel that there should be a much more straightforward way of seeing the result but can't seem to get there at the moment. Can someone help out?
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Answers (2)

Conor Daniel
Answered 2022-10-06 Author has 11 answers
You are right, that | | λ | | 2 = 1. With this information it is easy to see that
| | y y 2 λ | | 2 2 = 0.
To this end use: | | a | | 2 2 = ( a | a ), where ( | ) denotes the usual inner product.
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samuelaplc
Answered 2022-10-07 Author has 2 answers
A different approach, don't know if it's more straightforward, but maybe a bit more intuitive:
Take λ λ and complete it to an orthonormal basis { λ λ , e 2 , . . . , e n } . Then
y = λ , y λ λ 2 + i e i , y e i = y λ 2 λ + i e i , y e i .
Taking the norm of y and using λ < 1 , we see e i , y = 0 for i = 2 , . . . , n ; and also λ = 1. This yields
y = y λ .
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