Let Z ∈ R n , α ∈ R n , γ &gt; 0. Now I would like to find a solution for α of the following equation, Z − α = γ α ‖ α ‖ 2 . I am totally unfamiliar with such an equation. So I already get stuck here. Any help is really appreciated (or sources/references).

Question

Let   Z  ∈            R        n  ,   α  ∈            R        n  ,   γ  &amp;gt;  0. Now I would like to find a solution for   α of the following equation,                    Z        −        α        =        γ                  α                      ‖            α                          ‖                              2                                                    .            I am totally unfamiliar with such an equation. So I already get stuck here. Any help is really appreciated (or sources/references).

Bridger Hall · Accepted Answer

Z  −  α  =  γ      α          ‖      α              ‖                  2                      ⟺  Z  =  α      (    1    +    γ          1              ‖        α                  ‖          2                      )        (  ∗  )Taking the       |        |    .      |              |        2  , you get that      |        |    Z      |              |        2    =      |        |    α      |              |        2        (    1    +    γ          1              ‖        α                  ‖          2                      )    =      |        |    α      |              |        2    +  γso you must have       |        |    α      |              |        2    =      |        |    Z      |              |        2    −  γ, the equation has no solution. If       |        |    Z      |              |        2    &amp;gt;  γ, then   (  ∗  ) gives  Z  =  α      (    1    +    γ          1                        |                          |                Z                  |                                      |                    2                −        γ              )    ,    so    α  =      1          (      1      +      γ              1                              |                                |                    Z                      |                                              |                        2                    −          γ                    )        Z  =                    |                    |            Z              |                              |                2            −      γ                      |                    |            Z              |                              |                2              Z(if   Z  ≠  0) and you can check that this is indeed a solution.

Ignacio Casey · Accepted Answer

let             R        n    =  α      R    ⨁            (      α              R            )              ⊥      You have :   Z  =      z          α            u          α        +      z          ⊥            u          ⊥       and   α  =  α      u          α      So you get       z          ⊥        =  0 and :      z          α        =  α  +  γ    α  =      (          |              |        Z          |              |        −    γ    )        Z                  |                    |            Z              |                    |            (which is also equivalent to the result of the other answer)

Let Z in RR^n, alpha in RR^n, gamma>0. Now I would like to find a solution for alpha of the following equation, Z−alpha=gamma (alpha)/(norm(alpha)_2) I am totally unfamiliar with such an equation. So I already get stuck here. Any help is really appreciated (or sources/references).

Answered question

Answer & Explanation

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