Conditions on matrices such that $\| \boldsymbol{Ax} \|_2 \geq \| \boldsymbol{Bx} \|_2$ is satisfied

Question

Suppose we have two matrices:$\boldsymbol{A}$ with size $m_a {\times} n$, and $\boldsymbol{B}$ with size $m_b {\times} n$ such that $m_a > m_b$, and $m_a < n$. The entries of $\boldsymbol{A}$ and $\boldsymbol{B}$ take values in $\{0,1\}$.

Also, $\operatorname{rank}(\boldsymbol{A}) \geq \operatorname{rank}(\boldsymbol{B}) = m_b$.

Let $\boldsymbol{x}$ be an arbitrary vector in $\mathbb{C}^n$. I'm trying to find conditions on $\boldsymbol{A}$ and $\boldsymbol{B}$ such that \begin{equation} \| \boldsymbol{Ax} \|_2 \geq \| \boldsymbol{Bx} \|_2 \quad, \forall \boldsymbol{x} \in \mathbb{C}^n \end{equation}

I know that if the minimum singular value of $ \boldsymbol{A} $ is $\geq$ the maximum singular value of $\boldsymbol{B}$ then this inequality is satisfied. But this condition is rather too strict. I need to find milder conditions if they exist.

Also it would be of great help if you could provide general conditions for any matrices $\boldsymbol{C}$ and $\boldsymbol{D}$ with entries take values in $\mathbb{R}$ (or $\mathbb{C}$) such that

\begin{equation} \| \boldsymbol{Cx} \|_2 \geq \| \boldsymbol{Dx} \|_2 \quad, \forall \boldsymbol{x} \in \mathbb{C}^n \end{equation}

Thank you.

Answer 1

I figured out a condition on $\boldsymbol{C}$, $\boldsymbol{D}$ for the second part of the question.

We can find \begin{equation} || \boldsymbol {Cx} ||_2 \geq || \boldsymbol {Dx} ||_2 \quad ,\forall \boldsymbol{x} \in \mathbb{C}^n \end{equation} if and only if \begin{equation} \boldsymbol {C}^T \boldsymbol {C} - \boldsymbol {D}^T \boldsymbol {D} \succeq 0 \end{equation}

Proof:

We need to show that $\forall \boldsymbol{x} \in \mathbb{C}^n $, \begin{align} || \boldsymbol {Cx} ||_2 &\geq || \boldsymbol {Dx} ||_2 \\ \Longleftrightarrow || \boldsymbol {Cx} ||^2_2 &\geq || \boldsymbol {Dx} ||^2_2 \\ \Longleftrightarrow \left( \boldsymbol {Cx} \right)^T \left( \boldsymbol {Cx} \right) &\geq \left( \boldsymbol {Dx} \right)^T \left( \boldsymbol {Dx} \right) \\ \Longleftrightarrow \boldsymbol {x}^T \boldsymbol {C}^T \boldsymbol {C} \boldsymbol {x} &\geq \boldsymbol {x}^T \boldsymbol {D}^T \boldsymbol {D} \boldsymbol {x} \end{align} \begin{equation} \Longleftrightarrow \boldsymbol {x}^T \boldsymbol {C}^T \boldsymbol {C} \boldsymbol {x} - \boldsymbol {x}^T \boldsymbol {D}^T \boldsymbol {D} \boldsymbol {x} \geq 0 \end{equation} \begin{equation} \Longleftrightarrow \boldsymbol {x}^T \left( \boldsymbol {C}^T \boldsymbol {C} - \boldsymbol {D}^T \boldsymbol {D} \right) \boldsymbol {x} \geq 0 \end{equation} \begin{equation} \Longleftrightarrow \boldsymbol {C}^T \boldsymbol {C} - \boldsymbol {D}^T \boldsymbol {D} \succeq 0 \end{equation}