$ A = \begin{bmatrix} 1 & 1 & 1 \\ \epsilon & 0 & 0\\ 0 & \epsilon & 0 \\ 0 & 0 & \epsilon \end{bmatrix}$
Find $K_2(A)$ (where $K_2(A) $ represents the condition number of the matrix with respect to $\|.\|_2$ norm. )
I find $\|A\|_2 = \sqrt{\epsilon^3+3}$ but how find $\|A^{-1}\|_2=\|A^+\|_2$????
I find out that $A^+ =\frac{1}{\epsilon^4+3\epsilon^2}\begin{bmatrix} {\epsilon}^2+1 & \epsilon(\epsilon^2+2) & -\epsilon & -\epsilon \\ {\epsilon}^2+1 & -\epsilon & \epsilon(\epsilon^2+2) & -\epsilon\\ {\epsilon}^2+1 & -\epsilon & -\epsilon & \epsilon(\epsilon^2+2) \end{bmatrix} $
It's a book exercise problem. It's mentioned that this matrix is ill-conditioned .
