半酉矩阵相关量的无穷范数上界推导疑问

Hey there, let's break down how to bridge the gap between your derived bound and the one claimed in the paper. First, let's recap key identities and norm properties we'll need:

Key Setup & Identities #

• We know the orthogonal projection onto the complement of $\text{span}(U')$ is $U'\perp U'^T\perp = I - U'U'^T$, and similarly $V'\perp V'^T\perp = I - V'V'^T$.
• For any semi-unitary matrix $X$ (like $U$ or $V$), its operator norm $|X|_{op} = 1$, since $X^T X = I$.

Step 1: Rewrite $U'\perp U'^T\perp U$ using projection differences #

Let's expand the product $U'\perp U'^T\perp U$:
$$
U'\perp U'^T\perp U = (I - U'U'^T)U = U - U'U'^T U
$$
Notice that this is exactly the negative of $(U'U'^T - UU^T)U$:
$$
(U'U'^T - UU^T)U = U'U'^T U - UU^T U = U'U'^T U - U = - (U - U'U'^T U) = -U'\perp U'^T\perp U
$$
Taking the $2\to\infty$ norm of both sides gives:
$$
\left|U'\perp U'^T\perp U\right|{2\to\infty} = \left|(U'U'^T - UU^T)U\right|{2\to\infty}
$$

Step 2: Bound the $2\to\infty$ norm of the product #

Recall that for any matrices $A$ and $B$, the $2\to\infty$ norm satisfies $|AB|{2\to\infty} \leq |A|{2\to\infty} |B|{op}$. Since $U$ is semi-unitary, $|U|{op}=1$, so:
$$
\left|(U'U'^T - UU^T)U\right|{2\to\infty} \leq \left|U'U'^T - UU^T\right|{2\to\infty} \cdot |U|{op} = \left|U'U'^T - UU^T\right|{2\to\infty}
$$
Combining this with the previous equality gives:
$$
\left|U'\perp U'^T\perp U\right|{2\to\infty} \leq \left|U'U'^T - UU^T\right|{2\to\infty}
$$

Step 3: Repeat for the $V$-side term #

By exactly the same logic, we can show:
$$
\left|V'\perp V'^T\perp V\right|{2\to\infty} \leq \left|V'V'^T - VV^T\right|{2\to\infty}
$$

Step 4: Substitute back into your derived bound #

Now plug these two inequalities into the bound you already proved:
$$
\left|U'{\perp}\left(U^{{'T}_{\perp}MV'_{\perp}\right)V}{'T}{\perp}\right|{\infty} \leq \left|U'{\perp}U^{'T}{\perp}U\right|{2 \to \infty} \left|M\right|{op} \left|V'{\perp}V^{'T}{\perp}V\right|{2 \to \infty}
$$
Replacing each $2\to\infty$ norm term with the projection difference norm gives exactly the claim from the paper:
$$
\left|U'{\perp}\left(U^{{'T}_{\perp}MV'_{\perp}\right)V}{'T}{\perp}\right|{\infty} \leq \left|U'U'^T-UUT\right|{2 \to \infty} \left|M\right|{op} \left|V'V'^T-VVT\right|{2 \to \infty}
$$

备注：内容来源于stack exchange，提问作者Skywear