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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!

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# Maximise the following expression subject to the constraint

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