# Maximum singular value of matrix-valued function

Let $H_1$ and $H_2$ be two given $n\times n$ matrices. Consider the following matrix-valued function, which maps an arbitrary $n\times n$ unitary matrix $U$ to $$f(U)=U H_1 U^\dagger + i H_2$$ Given any $n\times n$ matrix $M$, define $\sigma_1(M)$ to be the greatest singular value of $M$.

Question: find $\underset{U}{\text{max}}\;\sigma_1(f(U))$.

• What is $U^{\dagger}$? – Arin Chaudhuri Jul 29 '17 at 23:11
• It denotes the Hermitian conjugate, i.e. the conjugate transpose of $U$. Mathematicians probably denote it more often by a star. – usr65465124 Jul 30 '17 at 17:50
• FWIW, I would advise you to add some context to your question. A question like this is very likely to be ignored or closed. – Arin Chaudhuri Jul 31 '17 at 15:12