I am trying to lower bound the following norm in the Euclidean space
\begin{align}\left \|Ax + By \right\|_{\infty},\end{align}where $A, B \in \mathbb{C}^{m \times n}$ and $x, y \in \mathbb{C}^n$.
I made an attempt, but I am not sure whether I am doing it correctly. Please correct me if I am wrong.
We can start with the definitions and properties:
- For any vector $v \in \mathbb{C}^n$, the infinity norm is defined as:\begin{align} \|v\|_{\infty} = \max_{i} |v_i| \end{align}
- For a matrix $M \in \mathbb{C}^{m \times n}$, the induced infinity norm is defined as:\begin{align} \|M\|_{\infty} = \max_{i} \sum_{j} |M_{ij}| \end{align}
- For any matrices $A, B \in \mathbb{C}^{m \times n}$ and vectors $x, y \in \mathbb{C}^n$, we have the property:\begin{align} \|Ax\|_{\infty} \leq \|A\|_{\infty} \|x\|_{\infty} \end{align}and similarly,\begin{align} \|By\|_{\infty} \leq \|B\|_{\infty} \|y\|_{\infty} \end{align}
The reverse triangle inequality for the infinity norm states:\begin{align}\|u + v\|_{\infty} \geq \left| \|u\|_{\infty} - \|v\|_{\infty} \right|\end{align}Applying this to $u = Ax$ and $v = By$, we get:\begin{align}\|Ax + By\|_{\infty} \geq \left| \|Ax\|_{\infty} - \|By\|_{\infty} \right|\end{align}
To find the lower bound, we need to understand how $\|Ax\|_{\infty}$ and $\|By\|_{\infty}$ relate to $\|A\|_{\infty}$ and $\|B\|_{\infty}$.
By the sub-multiplicative property of matrix norms, we have:\begin{align}\|Ax\|_{\infty} \leq \|A\|_{\infty} \|x\|_{\infty}\end{align}and\begin{align}\|By\|_{\infty} \leq \|B\|_{\infty} \|y\|_{\infty}\end{align}
Thus, we can write:\begin{align}\|Ax + By\|_{\infty} \geq \left| \|A\|_{\infty} \|x\|_{\infty} - \|B\|_{\infty} \|y\|_{\infty} \right|\end{align}
Combining these results, we can get the following bound for $\|Ax + By\|_{\infty}$:
\begin{align}\boxed{\left| \|A\|_{\infty} \|x\|_{\infty} - \|B\|_{\infty} \|y\|_{\infty} \right| \leq \left| \|x\|_{\infty} - \|y\|_{\infty} \right| \leq \|Ax + By\|_{\infty}},\end{align}where I assumed that $\|A\|_{\infty} \leq 1$ and $\|B\|_{\infty} \leq 1$.