I tried this exercise from the book numerical matrix analysis by llse. C. F. Ipsen of section 3.3 (iii). Let $A \in \mathbb{R}^{n \times n}$ be a matrix upper triangular and nonsingular. I have to show
$$\kappa_{\infty}(\boldsymbol{A}) \geq \frac{\|\boldsymbol{A}\|_{\infty}}{\min _{i \in\{1, \ldots, n\}}\left|a_{i i}\right|}$$
then i had the idea using the result for non-singular matrices where if $\boldsymbol{A}\in \mathbb{R}^{n \times n}$, then for all $\boldsymbol{B}\in \mathbb{R}^{n \times n}$ $$\kappa(\boldsymbol{A}) \geq \frac{\|\boldsymbol{A}\|}{||\boldsymbol{A-B}||}$$
and based on this fact then i would have to $$\kappa(\boldsymbol{A})_{\infty} \geq \frac{\|\boldsymbol{A}\|_{\infty}}{||\boldsymbol{A-B}||_{\infty}}$$
my question is am i ok?
Is sufficient to prove $||\boldsymbol{A-B}||_{\infty}=\min _{i \in\{1, \ldots, n\}}\left|a_{i i}\right|$ ? how could I prove the exercise? Any idea? Thanks.
You can benefit from a formula that describes the inverse of a nonsingular triangular matrix $T$. Let us suppose that $T$ is a nonsingular upper triangular matrix and let us write $$T = \left[ \begin{array}{c|c} t & u^T \\ \hline 0 & A \end{array} \right],$$ where $t$ is a nonzero scalar, $u^T$ is a row vector and $A$ is nonsingular upper triangular matrix. We either know or suspect that $T^{-1}$ is also an upper triangular matrix, hence we either know or conjecture that $T^{-1}$ can be written as $$ T^{-1} = \left[ \begin{array}{c|c} s & v^T \\ \hline 0 & B\end{array} \right]. $$ We now compute the product of the two matrices and find $$ \left[\begin{array}{c|c} t & u^T \\ \hline 0 & A \end{array} \right] \left[ \begin{array}{c|c} s & v^T \\ \hline 0 & B\end{array} \right] = \left[ \begin{array}{c|c} ts & tv^T + u^T B \\ \hline 0 & AB \end{array} \right]. $$ Obviously, we wish for the result to equal the identity matrix. This happens when $s=t^{-1}$, $B = A^{-1}$ and $v^T = -t^{-1} u^T A^{-1}$. We now make the critical observation that the diagonal elements of $T^{-1}$ are precisely the reciprocal values of the elements on the diagonal of $T$. Hence $$\forall i \: : \: \|T^{-1}\|_{\infty} \ge |(T^{-1})_{ii}| = |T_{ii}|^{-1}.$$ We conclude that $$\|T^{-1}\|_{\infty} \ge \frac{1}{\underset{i}{\min} \{ |T_{ii}| \} }.$$