Relations between matrix norms

Question

Let $A \in M_{n\times n}(\mathbb{C})$. Suppose $U_{1} \in M_{(n)\times (n-1)}$ such that columns of $U_{1}$ are orthogonal to each other. Consider the case

$$ B = {U_{1}}^{*} A U_{1}. $$

Can we say anything about the relationship between matrix norms $||A||$ and $||B||$ (can we say $||B|| \leq ||A||$)?

Plesae let me know if you had any idea.

Answer 1

First note that $||Uv||_p=||v||_p$ for any unitary matrix $U$ ($UU^H=U^HU=I$). Here we have $$\sup_{||x||_p=1}||Bx||_p{=\sup_{||x||_p=1}||U^HAUx||_p\\=\sup_{||Ux||_p=1}||U^HAUx||_p\\=\sup_{||y||_p=1}||U^HAy||_p\\=\sup_{||y||_p=1}||Ay||_p\\=||A||_p}$$therefore $$||B||_p=||A||_p$$