I want to find matrices $F$, $G$, and $H$ minimizing
$\begin{bmatrix} x^T & y^T& z^T \end{bmatrix} \begin{bmatrix} I & 0& 0 \\ 0 & F &0 \\ 0 & G &H \end{bmatrix}^{T} Q \begin{bmatrix} I & 0& 0 \\ 0 & F &0 \\ 0 & G &H \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} $
as a function of $x,y,z$ and symmetric positive definite matrix $Q$ given as
$\begin{bmatrix} Q_{11} & Q_{12}& Q_{13} \\ Q_{21} & Q_{22} &Q_{23} \\ Q_{31} & Q_{32} &Q_{33} \end{bmatrix}$ where $Q_{ij}$ matrices are with appropriate dimensions.
All $x,y,z$ are column vectors with the size of $n$ and $I$ denotes identity matrix. Can anybody give me some hints about how we can solve such a problem?
The problem can be written as
$\begin{bmatrix} x^T & y^T& z^T \end{bmatrix} \begin{bmatrix} Q_{11} & Q_{12}F + Q_{13}G& Q_{13}H \\ F^T Q_{21} + G^T Q_{31} & F^T Q_{22}F + F^T Q_{23} G + G^T Q_{32}F + G^T Q_{33} G & F^T Q_{23} H + G^T Q_{23} H \\ H^T Q_{31} & H^T Q_{32} F + H^T Q_{33} G & H^T Q_{33} G \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} $
Let $w = \begin{bmatrix} x^T \\ y^T\\ z^T \end{bmatrix}$ and $X = \begin{bmatrix} I & 0& 0 \\ 0 & F &0 \\ 0 & G &H \end{bmatrix}^T$, and you can write this as $\text{trace}((ww^T)XQX^T)$. Now study scalar-by-matrix identities and compare with $\text{trace}(AXBX^TC)$, and you have $A = ww^T, B = Q, C=I$. The derivative w.r.t $X$ is given by $B^TX^TA + BX^TA$. Let $S$ define the $0/1$ matrix defining which elements which actually are free to optimize in $X$ (locations corresponding to $F$,$G$,$H$), and you have to solve the linear system of equations $S \circ (B^TX^TA + BX^TA)=0$. For an even more explicit representation, you will have to rewrite this using various kronecker and vectorization tricks.