From related question
- highly non-normal linear system gives slow convergence for linear solvers. Is there geometric intuition why?
Figure in 7.2 of Corless/Fillion book :
Higham suggests using $\tau(A)$ the difference between squared singular value/eigenvalue magnitudes as a measure of non-normality of a matrix $A$:
$$\tau(A)=\sum_i |\sigma_i|^2 - |\lambda_i|^2$$
- Is there geometric intuition for what kinds of linear operations make this measure high?
Below is a simple test in $d=2$, combining stretching with a rotation. $$A(\theta)=\left( \begin{array}{cc} 10 & 0 \\ 0 & 1 \\ \end{array} \right)\left( \begin{array}{cc} \cos (\theta ) & -\sin (\theta ) \\ \sin (\theta ) & \cos (\theta ) \\ \end{array} \right)$$
It seems like deviation from normality is highest $(\tau=81)$ when strong scaling is combined with a strong rotation $(\theta\approx \pi/2)$. Can this intuition be extended to higher dimensions?
Here's what eigenvalues look like for various values of $\theta$
