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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
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