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 ≠ 0Also, ‖ 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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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    ≠  0Also,   ‖  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&#039;t seem to get there at the moment. Can someone help out?

Conor Daniel · Accepted Answer

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.

samuelaplc · Accepted Answer

A different approach, don&#039;t know if it&#039;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  ‖  &amp;lt;  1  , we see   ⟨      e    i    ,  y  ⟩  =  0 for   i  =  2  ,  .  .  .  ,  n  ; and also   ‖  λ  ‖  =  1. This yields  y  =  ‖  y  ‖  λ  .

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).

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