TITLE: Maximize $\text{trace}(Z^{T}A Z)/\text{trace}(Z^{T}B Z)$
QUESTION [0 upvotes]: I am trying to find $Z$ in the following problem:
\begin{equation}
 \begin{array}{c}
 maximize \hspace{2mm} \frac{\text{trace}(Z^{T}Lb\hspace{2mm} Z)}{\text{trace}(Z^{T}Lw\hspace{2mm} Z)}, \\
 \end{array}
\end{equation} for $Z \in \mathbb{R}^{m\times n}$ and $Wb,Ww \in \mathbb{R}^{mxm}$.
Using the scale constraint \begin{equation} Z^{T} Dw Z = I \end{equation}
And Lw = Dw −Ww , Lb = Db −Wb.(Laplacian)
W(Wb,Ww) denotes the symmetric affinity matrix and D(Db,Dw)
is the diagonal weight matrix, whose entries are column (or row, since W is
symmetric) sums of W, then the Laplacian matrix is given L = D −W.
I would appreciate any help!

REPLY [0 votes]: Hint:
See these links:
http://www-users.cs.umn.edu/~thango/reports/trace_ratio.pdf
http://users.cecs.anu.edu.au/~hongdong/Shen-ACCV2007.pdf
http://mmlab.ie.cuhk.edu.hk/archive/2007/CVPR07_reduction.pdf
http://www.fondation-hadamard.fr/sites/default/files/images/pgmo_2013_bellalij_2.pdf
You'll find some methods for solving the constrained problem of the ratio of traces.
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