E10.1.
Compute \(\mathrm{Cond}_1(A)\) and \(\mathrm{Cond}_\infty(A)\), the \(1\)- and \(\infty\)-norm condition numbers of \[A= \begin{bmatrix} 0.01 & 2\\ 0 & 1 \end{bmatrix}.\]
Compute \(\|A\|_2\), where \[A= \begin{bmatrix} 2 & 0 \\ 0 &1 \end{bmatrix}.\]
Let \(A\) be a diagonal matrix with entries \(d_1,d_2,\dots,d_n\). Give an expression for \(\mathrm{Cond}_2(A)\).
E10.2.
A scalar \(\lambda\) is an eigenvalue of a \(d\times d\) matrix \(A\) if there exists a vector \(\vec u\ne \vec 0\) such that \(A\vec u= \lambda\vec u\). Show that \(|\lambda| \leq \|A\|_{\mathrm{op}}\) for any operator norm and any eigenvalue \(\lambda\) of \(A\). condition number relative to an operator norm \(\|\cdot\|_{\mathrm{op}}\).
Show that \(\mathrm{Cond}(A)\ge 1\), where \(\mathrm{Cond}(\cdot)\) is the condition number relative to an operator norm \(\|\cdot\|_{\mathrm{op}}\).
An orthogonal matrix \(Q\) is a transformation that leaves the Euclidean distance constant, in other words \(\|Q\vec x\|_2 = \|\vec x\|_2\) for each vector \(\vec x\). Show that \(\|Q\|_{2} = 1\) for orthogonal \(Q\) and that \(\mathrm{Cond}_2(Q) = 1\). (Orthogonal matrices are perfectly conditioned.)
\(\star\)E10.3. Consider vectors \(\vec x\), \(\vec{\Delta x}\), \(\vec b\), and \(\vec{\Delta b}\) such that \[A \vec x= \vec b,\qquad A \Delta \vec x= \Delta \vec b.\]
By using \(\|A \vec x\| \le \|A\|_{\mathrm{op}} \, \|\vec x\|\), show that \[\frac{\|\vec{\Delta x}\|}{\|\vec x\|} \le \mathrm{Cond}(A) \frac{\|\vec{\Delta b}\|}{\|\vec b\|}.\]
By using the above, estimate \(\mathrm{Cond}(A)\) with respect to \(\|\cdot\|_\infty\) in the case \[A= \begin{bmatrix} 1 & 1/2 \\ 1/2 & 1/3 \end{bmatrix}, \qquad \vec{ \Delta x}= \begin{bmatrix} 0.4 \\ -0.9 \end{bmatrix}, \qquad \vec{ x}= \begin{bmatrix} -0.9 \\ -5 \end{bmatrix}.\]