Given a $p\times p$ square matrix $A$. Can I say that the 2 norm of their product is always bounded for any $p \times 1$ vector, please? That is,
$$ \| Ax \| <\infty, \forall x\in\mathbb R^p. $$
A book I am reading says something like the above. It sounds right. However, I am not sure whether it is correct since I did not see such claim before. Could anyone explain a bit, please? Thank you!
Page 4-6 through 4-8 in this document give a detailed derivation to prove the following: $$ \| Ax \| <\boldsymbol{K}\| x \|, \forall x\in\mathbb R^p $$ where, $\boldsymbol{K}=\sqrt{\sigma^2_1}$,
In other words $\boldsymbol{K}$ is the square root of the largest eigenvalue of $A\times A$ that will always be real-valued and non-negative. Hence, the above product is always bounded.