# how to derive the product of singular values of this matrix

Well, I know this matrix is too complex, but still any hints or ideas on working it out will be greatly appreciated.

To describe this problem, I have to define some notations beforehand.

1. Twsit: Call $\xi$ a twist if $$\xi = \begin{bmatrix} v \\ \omega\end{bmatrix}$$, where $\|\omega\|_2 = 1$ here and after.
2. hat transformation of $\omega$,$$\hat\omega = \begin{bmatrix}0 &-\omega_3 &\omega_2 \\ \omega_3& 0 & -\omega_1\\ -\omega_2 &\omega_1 & 0\end{bmatrix}$$ for 3-by-1 vector $\omega$
3. hat transformation of twist: define $\hat\xi = \begin{bmatrix}\hat\omega & v \\ 0& 0\end{bmatrix}$
4. Exponential Map of $\omega$: it can be proved that $$e^{\hat\omega\theta} = I_{3\times3} + \hat\omega\sin(\theta) + \hat\omega^2\left(1-\cos(\theta)\right)$$
5. Exponential Map of twist: also it's proved that $$e^{\hat\xi\theta}=\begin{bmatrix}e^{\hat\omega\theta}&(I-e^{\hat\omega\theta})(\omega\times{v})+\omega\omega^Tv\theta\\0_{1\times3}&1\end{bmatrix}$$
6. Adjoint transformation: define: $g = e^{\hat\xi\theta}=\begin{bmatrix}R&P\\0&1\end{bmatrix}$ the Adjoint transformation of $g$, $$Ad_g=\begin{bmatrix}R &\hat PR\\0&R\end{bmatrix}$$
7. because our problem involves n twists,for simplicity, let's denote: $$Ad_i^j=Ad_{e^{\hat\xi_1\theta_1^j}}Ad_{e^{\hat\xi_2\theta_2^j}}\cdots Ad_{e^{\hat\xi_i\theta_i^j}}$$ and specially, $Ad_0^j=I_{6\times6}$ where $\xi$-s are twists and $\theta$ are scalars.

Fianlly we can begin my problem. I came across a matrix of the following form $$\it{O}=\begin{bmatrix} A_1\\A_2\\\vdots\\A_m \end{bmatrix}$$ with $$A_j=\begin{bmatrix} I-Ad_1^j& Ad_1^j-Ad_2^j&\cdots&Ad_{n-1}^j-Ad_n^j&Ad_n^j\\ \end{bmatrix}$$ let $\sigma_1,\sigma_2,\cdots,\sigma_L$ be non-zero singular values of $\it{O}$ The problem is how can I choose some combinations of $\theta$-s to maximize the product of all its singular values?

Problem : to maximize $\prod\limits_{i=1}^{L}$ by selecting proper $\theta$-s

Frankly speaking, I don't think close form solution would exist, but is there any way for an iterative algorithm to exist? Analytic solution, algorithm or any hints is welcome.