Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Maximise the following expression subject to the constraint

  1. Jul 9, 2008 #1
    Hi guys,

    I wish to maximise the following expression subject to the constraint that [itex]\|\underline{w}\| = 1[/itex], and [itex]\mathbb{R}[/itex] is fixed.

    [tex]
    P = \underline{w}^H \mathbb{G}^H\mathbb{G}\underline{w}
    = \underline{w}^H \mathbb{R} \underline{w}
    [/tex]

    where

    [tex]
    \mathbb{R} \triangleq \mathbb{G}^H\mathbb{G}
    [/tex]

    I proceed to determine the maximum value, and the value of [itex]\underline{w}[/itex] that achieves it through the general eigenvalue problem and the Cauchy-Schwarz inequality.

    Recall that I can only modify w, and its norm is fixed to unity.

    [tex]
    |\langle \underline{w},\mathbb{R}\underline{w}\rangle| \leq \|\underline{w}\|\|\mathbb{R}\underline{w}\|
    [/tex]

    This achieves equality iff

    [tex]
    \mathbb{R}\underline{w} = \lambda \underline{w}
    [/tex]

    and will be subsequently maximised if the dominant eigenvector is chosen, such that

    [tex]
    \mathbb{R}\underline{w} = \lambda_{max} \underline{w}
    [/tex]

    Which then yields a maximum value of [itex]P[/itex] as

    [tex]
    P = \lambda_{max} \|w\|^2 = \lambda_{max}
    [/tex]

    I just want to doubly check with you guys that this is correct.

    Thanks!
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you help with the solution or looking for help too?
Draft saved Draft deleted



Similar Discussions: Maximise the following expression subject to the constraint
Loading...