Let $A$ be a nonsingular $n \times n$ matrix, $\|\cdot\|$ be any natural norm, and $K_{p}(A)=$ $\|A\|_{p}\left\|A^{-1}\right\|_{p} .$ Let $\lambda_{1}$ be the smallest and $\lambda_{n}$ be the largest eigenvalues of the matrix $A^{t} A$
(a) Show that if $\lambda$ is an eigenvalue of $A^{t} A$, then $0<\lambda \leq\left\|A^{t}\right\|\|A\|$.
(b) Show that $K_{2}(A)=\sqrt{\frac{\lambda_{n}}{\lambda_{1}}}$.
(c) Show that $K_{2}(A) \leq \sqrt{K_{1}(A) K_{\infty}(A)} . \quad$ (As a definition, $\|A\|_{1}=\max _{1 \leq j \leq n} \sum_{i=1}^{n}\left|a_{i j}\right|$ and $\|A\|_{\infty}=$
$\left.\max _{1 \leq i \leq n} \sum_{j=1}^{n}\left|a_{i j}\right| .\right)$
$(a)$ and $(b)$ are straightforwardly easy from the statements; however, $(c)$ seems to be a little ambiguous due to the norm of matrix $A^{-1}$. Are there any ideas to solve the inequality more mathematically?
Best Answer
This follows directly from the inequalty that $$ \|A\|_2^2=\rho(A^\ast A)\le\|A^\ast A\|_\infty\le\|A^\ast\|_\infty\|A\|_\infty=\|A\|_1\|A\|_\infty. $$