Maximisation

[brendan_foo]

Hi guys,

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


P = \underline{w}^H \mathbb{G}^H\mathbb{G}\underline{w}
= \underline{w}^H \mathbb{R} \underline{w}


where


\mathbb{R} \triangleq \mathbb{G}^H\mathbb{G}


I proceed to determine the maximum value, and the value of \underline{w} 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.


|\langle \underline{w},\mathbb{R}\underline{w}\rangle| \leq \|\underline{w}\|\|\mathbb{R}\underline{w}\|


This achieves equality iff


\mathbb{R}\underline{w} = \lambda \underline{w}


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


\mathbb{R}\underline{w} = \lambda_{max} \underline{w}


Which then yields a maximum value of P as


P = \lambda_{max} \|w\|^2 = \lambda_{max}


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

Thanks!