# Maximizing the matrix norm

Hi all,

I wish to find a 3x3 rotation (orthogonal) matrix $\mathbf{R}$ such that it maximizes the following matrix norm:

$||\mathbf{A} \mathbf{D}_r \mathbf{D}_b ||_2$

where $\mathbf{A}$ is a known $M \times 3N$ matrix,

$\mathbf{D}_r = diag(\mathbf{R})$ is the unknown $3N \times 3N$ block diagonal matrix,

$\mathbf{D}_b = diag(\mathbf{v}_i), i=1\dots N$ is a known $3N \times N$ block diagonal matrix.

This optimization problem arises from an engineering estimation problem with noisy measurements. I believe it is way beyond a trivial math problem, at least for an engineer like me. I hope that someone can give me some advice on how to tackle it. Thanks so much.

Daniel

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Just find the critical points of your expression by differentiating it, keeping in mind the constraint $R^tR = I$. – Deane Yang Mar 6 '11 at 16:54
Deane Yang: Could you please elaborate more? How can I differentiate an induced norm? Am I missing something? If possible, any reference to materials, papers is greatly appreciated. – Danny Kane Mar 7 '11 at 6:51