Problem
I am wondering if there is a way to efficiently compute the maximum singular value of \begin{align} C(\phi)=A \cos\phi+B \sin\phi, \end{align} where A, B and C are real 2x2 matrices.
Numerical Approach (edited)
So far, I could only solve the problem 'numerically' by
- Direct optimization (e.g. using matlabs's
fmincon), which is sensitive to the choice of the intial value of $\phi$. Other (global) optimizers are more reliable but need far more evaluations. Maybe an appropriate bisection algorithm would help here? Suggestions? - Rank-1 tensor approximation (e.g. using 'tenser toolbox' for matlab), which also needs quite some iterations. I am not very familiar with tensor algebra so I am not sure if this approach is even valid in general.
Notes (edited)
- It can be assumed that either $A$ or $B$ is positive semidefinite.
- QZ-decompositon may be applied to make $A$ and $B$ (and thus also $C$) triangular. But I am not sure if this helps.
In the special case of $\phi=\pi/4$, you're essentially asking for the maximum singular value of $A+B$. So without further structural assumptions imposed on $A$ and $B$, the question is basically unanswerable, as the eigenvalues of $A+B$ can basically be anything within the spectrum of $A$ and $B$.