# Optimization with a term related to its singular values?

How can we find the maximum value of $$\left( \vec{k}^TT_1\vec{m} \right) \left( \vec{n}^TT_2\vec{l} \right) +\left( \vec{k}_{\bot}^{T}T_1\vec{m}_{\bot} \right) \left( \vec{n}_{\bot}^{T}T_2\vec{l}_{\bot} \right)$$ with $$k$$, $$l$$, $$m$$ and $$n$$ changable unit vectors, notation $$\bot$$ stands for perpendicular, and $$T_1$$ and $$T_2$$ are two 3 by 3 matrices? This can be formally stated in formula as $$\underset{k,l,m,n}{\max}\left( \vec{k}^TT_1\vec{m} \right) \left( \vec{n}^TT_2\vec{l} \right) +\left( \vec{k}_{\bot}^{T}T_1\vec{m}_{\bot} \right) \left( \vec{n}_{\bot}^{T}T_2\vec{l}_{\bot} \right)$$.

Here are some of my thinkings: I think that we can use singular value decomposition of $$T_1$$ and $$T_2$$, then by changeable of unit vectors, we can take $$T_1$$ and $$T_2$$ as diagonal vectors. Then I guess the maximum value is $$\xi _1\eta _1+\xi _2\eta _2$$ where $$\xi_1\ge\xi_2\ge\xi_3$$ are singular values of matrix $$T_1$$, and the same meaning for $$\eta_i$$, with $$i=1,2,3$$. I did an intense numerical search by generating random vectors to verify this. But I don't know how analytically prove it. I find that when $$\vec m$$ stands for unit vector set, i.e. unit sphere, then $$T_1 \vec m$$ is the set of an ellipsoid, but the hard thing for me to prove it geometrically is that I found $$T_1 \vec m$$ no longer perpendicular to $$T_1 \vec m_\bot$$ all the time, or else we can take the maximum value of term $$\vec{k}^TT_1\vec{m}$$ simply as the length of the vector after the action of $$T_1$$.