Consider the following function of unitary matrices ${\bf U},{\bf V} \in\mathbb{R}^{N\times N}$: $$f({\bf U},{\bf V}) = \sigma_\mathrm{max}({\bf AUBVC})$$ whereby $\sigma_\mathrm{max}$ denotes the largest singular value. Everything is real-valued. ${\bf A}\in\mathbb{R}^{M\times N},{\bf B}\in\mathbb{R}^{N\times N},{\bf C}\in\mathbb{R}^{N\times O}$ are diagonal matrices with entries in descending order (i.e. the top left element is largest).
I am concerned with the following questions:
- [the easy one] It is quite clear that $$\max_{{\bf U},{\bf V}} f({\bf U},{\bf V}) = |A_{1,1} \, B_{1,1} \, C_{1,1}|.$$ What is the most concise way to prove this rigorously?
- [the hard one] Regarding the minimum value $$\min_{{\bf U},{\bf V}} f({\bf U},{\bf V})$$ can we state a closed-form expression? If not, can we state a good approximation or bound?
Thank you.