I have $y = Ax$ where x, y are vectors and $A$ is a matrix. I want to get the best $K$ such that $||y|| \leq K||x||$. Ideally, $K$ is a matrix norm. Especially, $K$ can be a subordinate matrix norm. I tried with Spectral norm and Frobenius norms and they provide a very loose bound for $K$. Are there studies in this direction?
I have also tried the largest sigular value of A and that is a worse bound too.
If that helps in my case A can be expressed as $(I-BC)^{-1}$ where $B$ and $C$ are symmetric metrices.
Let $\lVert A \rVert_\epsilon := \lVert D^{-1} T^{-1} A T D \rVert_\infty$ where $D := \operatorname{diag}\{\delta, \delta^2, \dots, \delta^n\}$ for some $1 > \delta > 0$ and $T$. It is easy to check that $\lVert \cdot \rVert_\epsilon$ is a submultiplicative norm.
Let $T$ be such that $T^{-1} A T = \Lambda + U$ where $\Lambda$ is diagonal and $U$ is strictly upper triangular. Therefore, $$ \lVert A \rVert_\epsilon = \lVert \Lambda + D^{-1} U D \rVert_\infty \leq \rho(A) + \lVert D^{-1} U D \rVert_\infty $$
where $\rho(\cdot)$ is the spectral radius. Note that $(D^{-1} U D)_{ij} = \delta^{j-i} U_{ij}, \forall j > i$. Therefore, $$ \lVert D^{-1} U D \rVert_\infty = \max_i \sum_{j=1}^n \lvert \delta^{j-i} U_{ij} \rvert \leq \delta \max_i \sum_{j=1}^n \lvert U_{ij} \rvert = \delta \lVert U \rVert_\infty $$
By choosing $\delta := \epsilon / \lVert U \rVert_\infty$ for some $\epsilon > 0$, we can obtain $$ \lVert A \rVert_\epsilon \leq \rho(A) + \epsilon $$
Now you can make $\epsilon$ as small as you like.