Bounds on complex vector equations using the 2-norm

Question

Let $\boldsymbol{x}$ and $\boldsymbol{y}$ be vectors of size $n {\times} 1$ and $m {\times} 1$, respectively, and let $\boldsymbol{A}$ be a matrix of size $m {\times} n$. Suppose $\boldsymbol{x}$, $\boldsymbol{y}$ and $\boldsymbol{A}$ are over the complex field $\mathbb{C}$.

I have this simple vector equation: \begin{equation} \boldsymbol{y} = \boldsymbol{A} \boldsymbol{x}. \end{equation}

I need to find a bound on $|| \boldsymbol{y} ||_2$ in the form of: \begin{equation} || \boldsymbol{y} ||_2 \ge \alpha || \boldsymbol{x} ||_2, \end{equation} where $\alpha > 0$ is a constant and $||\:.||_2$ is the 2-norm.

Any help is greatly appreciated. Thank you!

Note: I did find a bound with the opposite inequality sign, specifically, $|| \boldsymbol{y} ||_2 \le ||\boldsymbol{A}|| \: || \boldsymbol{x} ||_2$, where $||\boldsymbol{A}||$:= $\sup_{||\boldsymbol{x}||_2 \le 1} \{||\boldsymbol{Ax}||_2\}$.

Answer 1

Let $\underline{\sigma} = \min_{\|x\|_2 = 1} \|Ax\|_2$.

Note that $\underline{\sigma} > 0$ iff $\ker A = \{0\}$ (this follows from compactness of the unit ball).

$\underline{\sigma}$ is the smallest singular value of $A$.

A standard reference (overkill in this instance) would be "Matrix Computations" by Gene Golub, Charles Van Loan & Justin Bieber. I am not sure about the last author.